Solved: Find an explicit formula for and use it to graph,

(a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one?

(a) Suppose is a one-to-one function with domain and range . How is the inverse function defined? What is the domain of ? What is the range of ? (b) If you are given a formula for , how do you find a formula for ? (c) If you are given the graph of , how do you find the graph of ? f

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.f(x) = x2 - 2x

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.f(x) = 10 - 3x

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.g (x )= 1/x

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.g (x) = cos x

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.f (t) is the height of a football t seconds after kickoff.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.f (t) is your height at the age t.

If is a one-to-one function such that , what is ?

If , find .

If , find .

The graph of is given. (a) Why is one-to-one? (b) What are the domain and range of ? (c) What is the value of ? (d) Estimate the value of .

The formula , where , expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

In the theory of relativity, the mass of a particle with speed is where is the rest mass of the particle and is the speed of light in a vacuum. Find the inverse function of and explain its meaning.

Find a formula for the inverse of the function.f x 3 2x 22.

Find a formula for the inverse of the function.fx 4x 1 2x 3 f x 3

Find a formula for the inverse of the function.fx s10 3x 3 3

Find a formula for the inverse of the function.y 2x 5. 6. fx s10 3x 3 3 7. 9.

Find a formula for the inverse of the function.y 2 8x 1 sx 1 sx y

Find a formula for the inverse of the function.f x 2x x  2 y 2

Find an explicit formula for and use it to graph, and the line on the same screen. To check your work, see whether the graphs of and are reflections about the line.f x x x  0 2

Find an explicit formula for and use it to graph, and the line on the same screen. To check your work, see whether the graphs of and are reflections about the linef x sx x  0 f x

Use the given graph of to sketch the graph of .

Use the given graph of to sketch the graph of .

(a) Show that is one-to-one. (b) Use Theorem 7 to find . (c) Calculate and state the domain and range of . (d) Calculate from the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of and on the same axes. f x x a 8 3f 1

(a) Show that is one-to-one. (b) Use Theorem 7 to find . (c) Calculate and state the domain and range of . (d) Calculate from the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of and on the same axes. fx sx 2 a 2 f x

(a) Show that is one-to-one. (b) Use Theorem 7 to find . (c) Calculate and state the domain and range of . (d) Calculate from the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of and on the same axes. f x 9 x 0 x 3 a 8 2 fx s

(a) Show that is one-to-one. (b) Use Theorem 7 to find . (c) Calculate and state the domain and range of . (d) Calculate from the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of and on the same axes. f x 1x 1 x  1 a 2 f x 9

Find .f x x a 1 3 x 1 f 1

Find .f x x a 2 5 x 3 2x f x x

Find .f x 3 x 1  x  1 a 3 37. 2 tanx2 f x x a

Find .f x sx a 2 3 x 2 x 1 f x 3

Suppose is the inverse function of a differentiable function and . Find .

Suppose is the inverse function of a differentiable function and let . If and , find .

Show that the function is oneto-one. Use a computer algebra system to find an explicit expression for . (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

Show that , , is not one-to-one, but its restriction , , is one-to-one. Compute the derivative of by the method of Note 2.

(a) If we shift a curve to the left, what happens to its reflection about the line ? In view of this geometric principle, find an expression for the inverse of tx f x c, where is a one-to-one function. f y (b) Find an expression for the inverse of , where .

(a) If is a one-to-one, twice differentiable function with inverse function , show that (b) Deduce that if is increasing and concave upward, then its inverse function is concave downward.

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figure 4 and, if necessary, the transformations of Section 1.2. y ln  x  y

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figure 4 and, if necessary, the transformations of Section 1.2. y lnx 3 y 1

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figure 4 and, if necessary, the transformations of Section 1.2. y 1 lnx 2 y l

Differentiate the function.fx sx ln x 2

Differentiate the function.fx lnx fx sx ln x 2 10 y lnx

Differentiate the function.f lncos  fx c

Differentiate the function.fx cosln x fx

Differentiate the function.f x s x 5 ln x f

Differentiate the function.f x ln s 5 f x s x 5 ln x

Differentiate the function.tx ln 2 1) a x a x 19. f

Differentiate the function.hx ln(x sx tx ln 2 1) a x a

Differentiate the function.fu ln u 1 ln2u hx l

Differentiate the function.ft 1 ln t 1 ln t fu ln

Differentiate the function.Ft ln x 2t 1 3 3t 1 4 ft 1 ln

Differentiate the function.lnx4 sin2 Ft ln x 2t 1

Differentiate the function.y ln  2 x 5x2  y

Differentiate the function.Gu ln 3u 2 3u 2 y ln

Differentiate the function.y ln x 1 x 1  35 Gu

Differentiate the function.y ln tan x 2 y

Differentiate the function.y tan lnax b y

Differentiate the function.y ln  tan 2x

Find and . y ln ln x

Find and . y ln x x 2 y

Differentiate and find the domain of . fx fx ln ln ln x x 1 lnx 1 33. f f y

Differentiate and find the domain of . fx ln ln ln x x

If , find .

If , find .

Find an equation of the tangent line to the curve at the given point y sin2 ln x 1, 0 3 7,

Find an equation of the tangent line to the curve at the given point y lnx 37. y sin2 ln x 1, 0 3 7, 2, 0 ft t ln4

Find if .

Find if .

Find a formula for if .

Find .

Use a graph to estimate the roots of the equation correct to one decimal place. Then use these estimates as the initial approximations in Newtons method to find the roots correct to six decimal places. x 4 x 2 ln x d9 d

Use a graph to estimate the roots of the equation correct to one decimal place. Then use these estimates as the initial approximations in Newtons method to find the roots correct to six decimal places. ln4 x 2 x 4 x 2 ln x

Discuss the curve under the guidelines of Section 3.4. y lnsin x x

Discuss the curve under the guidelines of Section 3.4. y lntan2 y lnsin x x ln4

Discuss the curve under the guidelines of Section 3.4. y ln1 x 2 3x 2 2 y lnt

Discuss the curve under the guidelines of Section 3.4. y lnx y ln1 x 2 3x 2 2 y l

If , use the graphs of , , and to estimate the intervals of increase and the inflection points of on the interval .

Investigate the family of curves . What happens to the inflection points and asymptotes as changes? Graph several members of the family to illustrate what you discover.

Use logarithmic differentiation to find the derivative of the function. y 2x 1 5 x 4 3 6 c f x

Use logarithmic differentiation to find the derivative of the function.y x 3 1 4 sin2 x s 3 x y 2

Use logarithmic differentiation to find the derivative of the function.y sin2 x tan4 x x 2 1 2 y

Use logarithmic differentiation to find the derivative of the function.y 4 x 2 1 x 2 1 y

Evaluate the integral.y du 2 1 dt 8 3t

Evaluate the integral.y 2 1 4 u2 u3 y du 2

Evaluate the integral.y dx e 1 x 2 x 1 x dx y

Evaluate the integral.y 9 4 sx 1 sx  2 y dx

Evaluate the integral.y 2 x 2 6x x 3 dx 6

Evaluate the integral.y 6 e dx x ln x

Evaluate the integral.y dx ln x 2 x 61. dx y

Evaluate the integral.y cos x 2 sin x y dx l

Show that by (a) differentiating the right side of the equation and (b) using the method of Example 12.

Find if , , , and .

If is the inverse function of , find .

(a) Find the linear approximation to near l. (b) Illustrate part (a) by graphing and its linearization. (c) For what values of is the linear approximation accurate to within 0.1?

(a) By comparing areas, show that (b) Use the Midpoint Rule with to estimate .

Refer to Example 1. (a) Find an equation of the tangent line to the curve that is parallel to the secant line . (b) Use part (a) to show that .

By comparing areas, show that 2 1 3 1 n  ln n  1 1 2 1 3 1 n 1 262 C

Prove the third law of logarithms. [Hint: Start by showing that both sides of the equation have the same derivative.]

For what values of do the line and the curve enclose a region? Find the area of the region.

Find .

Use the definition of derivative to prove that lim xl0 ln1 x x 1 73.

(a) Compare the rates of growth of and by graphing both and in several viewing rectangles. When does the graph of finally surpass the graph of ? (b) Graph the function in a viewing rectangle that displays the behavior of the function as . (c) Find a number such that whenever x  N

(a) How is the number defined? (b) What is an approximate value for ? (c) Sketch, by hand, the graph of the function with particular attention to how the graph crosses the yaxis. What fact allows you to do this?

Solve each inequality for . e ln x  1 x  10 x

Solve each inequality for . e 2  ln x  9 23x  4 e

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graph given in Figure 2 and, if necessary, the transformations of Section 1.3. y ex e

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graph given in Figure 2 and, if necessary, the transformations of Section 1.3. y 1 2ex y

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graph given in Figure 2 and, if necessary, the transformations of Section 1.3.. y 3 e x 13.

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graph given in Figure 2 and, if necessary, the transformations of Section 1.3. y 2 51 e x y 3 e x 13. y

Find the limit. lim xl e1 x 3

Find the limit. lim xl2etan x lim

Find the limit. im x l e 3x e3x e 3x e3x lim

Find the limit. lim x l e 3x e3x e 3x e3x lim

Find the limit. lim xl2 e 32x lim

Find the limit. im xl2 e 32x lim

Differentiate the function. \f x x 2 ex li

Differentiate the function. y ex 1 x f

Differentiate the function. y e cos u cu ax3 23

Differentiate the function. y e u y e cos u cu ax3

Differentiate the function. u e ln x 1/u y

Differentiate the function. y ex fu e ln x 1/u

Differentiate the function. Ft et sin 2t 27.

Differentiate the function. y ek tan sx Ft

Differentiate the function. y s1 2e 3x y

Differentiate the function. y cosex y s1 2e 3x y

Differentiate the function. y ee x

Differentiate the function. y s1 xe2x y

Differentiate the function. y aex b cex d y

Differentiate the function. y ex ex e x ex y ae

Find an equation of the tangent line to the curve at the given point. y e 36. x, 1, e 2x cos x, 0, 1 y e

Find an equation of the tangent line to the curve at the given point. y ex y e 36. x, 1, e 2x

Find if .

Show that the function satisfies the differential equation .

For what values of does the function satisfy the equation ?

Find the values of for which satisfies the equation .

If , find a formula for .

Find the thousandth derivative of .

(a) Use the Intermediate Value Theorem to show that there is a root of the equation . (b) Use Newtons method to find the root of the equation in part (a) correct to six decimal places.

Use a graph to find an initial approximation (to one decimal place) to the root of the equation Then use Newtons method to find the root correct to six decimal places.

Under certain circumstances a rumor spreads according to the equation where is the proportion of the population that knows the rumor at time and and are positive constants. (a) Find . (b) Find the rate of spread of the rumor. ; (c) Graph for the case , with measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor.

An object is attached to the end of a vibrating spring and its displacement from its equilibrium position is , where is measured in seconds and is measured in centimeters. (a) Graph the displacement function together with the functions and . How are these graphs related? Can you explain why? (b) Use the graph to estimate the maximum value of the displacement. Does it occur when the graph touches the graph of ? (c) What is the velocity of the object when it first returns to its equilibrium position? (d) Use the graph to estimate the time after which the displacement is no more than 2 cm from equilibrium.

Find the absolute maximum value of the function .

Find the absolute minimum value of the function , .

On what interval is the curve concave upward?

On what interval is the function increasing?

Discuss the curve using the guidelines of Section 3.4.y 11 e x 51. 39.

Discuss the curve using the guidelines of Section 3.4.y e2 x ex y

Discuss the curve using the guidelines of Section 3.4.y e3x e2x y

Draw a graph of that shows all the important aspects of the curve. Estimate the local maximum and minimum values and then use calculus to find these values exactly. Use a graph of to estimate the inflection points. f x ecos x f

Draw a graph of that shows all the important aspects of the curve. Estimate the local maximum and minimum values and then use calculus to find these values exactly. Use a graph of to estimate the inflection points. f x ex 3 x f x

The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, lets scale the function so as to remove the factor and lets analyze the special case where . So we study the function (a) Find the asymptote, maximum value, and inflection points of . (b) What role does play in the shape of the curve? ; (c) Illustrate by graphing four members of this family on the same screen.

Evaluate the integral.5 0 e3x dx

Evaluate the integral.y 1 0 xex2 y dx 5

Evaluate the integral.e dx x s1 ex 59. dx

Evaluate the integral.sec2 x etan x y e dx x

Evaluate the integral.y dx ex 1 e x dx y

Evaluate the integral.y e 1x x 2 y dx e

Evaluate the integral.esx sx 63. dx

Evaluate the integral.e x sinex y dx es

Find the inverse function of . Check your answer by graphing both and on the same screen. f x lnx 3 f 1 f

Find the inverse function of . Check your answer by graphing both and on the same screen. f x 1 ex 1 ex f x

If , find .

Evaluate .

Prove the second law of exponents [see (7)].

Prove the third law of exponents [see (7)].

(a) Show that if . [Hint: Show that is increasing for .] (b) Deduce that .

(a) Use the inequality of Exercise 71(a) to show that, for (b) Use part (a) to improve the estimate of given in Exercise 71(b).

(a) Use mathematical induction to prove that for and any positive integer , ex  1 x x 2 2! x n n! n (b) Use part (a) to show that . (c) Use part (a) to show that for any positive integer .

This exercise illustrates Exercise 73(c) for the case . (a) Compare the rates of growth of and by graphing both and in several viewing rectangles. When does the graph of finally surpass the graph of ? (b) Find a viewing rectangle that shows how the function behaves for large . (c) Find a number such that whenever x  N

(a) Write an equation that defines when is a positive number and is a real number. (b) What is the domain of the function ? (c) If , what is the range of this function? (d) Sketch the general shape of the graph of the exponential function for each of the following cases. (i) (ii) (iii)

(a) If is a positive number and , how is defined? (b) What is the domain of the function ? (c) What is the range of this function? (d) If , sketch the general shapes of the graphs of and with a common set of axes.

Write the expression as a power of e 5s7

Write the expression as a power of e 10x2

Write the expression as a power of e cos x x 1

Write the expression as a power of e x cos x

Evaluate the expression. 16 log10 1000

Evaluate the expression. log10 0.1 log8 320 log8 5 l

Evaluate the expression. log5 5s2 log12 3 log12 48 l

Evaluate the expression. 10log10 4log10 7 loga 1

Graph the given functions on a common screen. How are these graphs related? y 20x y 5x y ex y 2x 10lo

Graph the given functions on a common screen. How are these graphs related? y ( 1 10 ) x y ( 1 3 ) x y 10x y 3x

Use Formula 6 to evaluate each logarithm correct to six decimal places. log12 e log6 13.54 log2 

Use Formula 6 to graph the given functions on a common screen. How are these graphs related? y log2 x y log4 x y log6 x y log8 x log1

Use Formula 6 to graph the given functions on a common screen. How are these graphs related? y log1.5 x y ln x y log10 x y log50 x y l

Use Formula 6 to graph the given functions on a common screen. How are these graphs related? y 10x y ex y ln x y log10 x y l

Find the exponential function whose graph is given.

Find the exponential function whose graph is given.

(a) Show that if the graphs of and are drawn on a coordinate grid where the unit of measurement is 1 inch, then at a distance 2 ft to the right of the origin the height of the graph of is 48 ft but the height of the graph of is about 265 mi. (b) Suppose that the graph of is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches ft?

Compare the rates of growth of the functions and by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place.

Find the limit.lim 2 5x 6 tl 2t 2 tx

Find the limit.lim xl3 log10x lim 2 5x 6 tl

Differentiate the function.ht t 3 3t lim

Differentiate the function.tx x4 4x ht

Differentiate the function.y 51x t

Differentiate the function.y 10tan

Differentiate the function.fu 2u 2u 10 27. y

Differentiate the function.y 23 x 2

Differentiate the function.y xx 31

Differentiate the function.f x log10 x x 1 f x log3x  2 4 y

Differentiate the function.y xx 31

Differentiate the function.y x 1x y

Differentiate the function.y xsin x 3

Differentiate the function.y sin x x y

Differentiate the function.y ln x x y

Differentiate the function.y xln x y

Differentiate the function.y xe x

Differentiate the function.y ln x cos x y

Find an equation of the tangent line to the curve at the point .

If , find . Check that your answer is reasonable by comparing the graphs of and .

Evaluate the integral. y du 2 1 10t dt

Evaluate the integral. y 1 0 42u y du 2

Evaluate the integral. y dx log10 x x dx

Evaluate the integral. y x 5 5x y dx log

Evaluate the integral. y 3 dx sin 45. cos  d

Evaluate the integral. y 2x 2x 1 y 3 dx si

Find the inverse function of .

Find if .

Calculate .

According to the Beer-Lambert Law, the light intensity at a depth of meters below the surface of the ocean is where is the light intensity at the surface and is a constant such that . (a) Express the rate of change of with respect to in terms of . (b) If and , find the rate of change of intensity with respect to depth at a depth of 20 m. (c) Using the values from part (b), find the average light

rove the second law of exponents [see (3)].

Prove the fourth law of exponents [see (3)].

Deduce the following laws of logarithms from (3): logaxy loga x loga y I0 logaxy loga x loga y loga logax y y loga x lo

Show that for any .

A population of protozoa develops with a constant relative 3. growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.

A population of protozoa develops with a constant relative 3. growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. (a) Find an expression for the number of bacteria after hours. (b) Find the number of bacteria after 3 hours. (c) Find the rate of growth after 3 hours. (d) When will the population reach 10,000?

A bacteria culture grows with constant relative growth rate. After 2 hours there are 600 bacteria and after 8 hours the count is 75,000. (a) Find the initial population. (b) Find an expression for the population after hours. (c) Find the number of cells after 5 hours. (d) Find the rate of growth after 5 hours. (e) When will the population reach 200,000?

The table gives estimates of the world population, in millions, from 1750 to 2000: (a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population. (c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.

The table gives the population of the United States, from census figures in millions, for the years 19002000. (a) Use the exponential model and the census figures for 1900 and 1910 to predict the population in 2000. Compare with the actual figure and try to explain the discrepancy. (b) Use the exponential model and the census figures for 1980 and 1990 to predict the population in 2000. Compare with the actual population. Then use this model to predict the population in the years 2010 and 2020. ; (c) Graph both of the exponential functions in parts (a) and (b) together with a plot of the actual population. Are these models reasonable ones?

Experiments show that if the chemical reaction takes place at , the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: (a) Find an expression for the concentration N O after t seconds if the initial concentration is . (b) How long will the reaction take to reduce the concentration of N O to 90% of its original value?

Bismuth-210 has a half-life of 5.0 days. (a) A sample originally has a mass of 800 mg. Find a formula for the mass remaining after days. (b) Find the mass remaining after 30 days. (c) When is the mass reduced to 1 mg? (d) Sketch the graph of the mass function.

The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after years. (b) How much of the sample remains after 100 years? (c) After how long will only 1 mg remain?

A sample of tritium-3 decayed to 94.5% of its original amount after a year. (a) What is the half-life of tritium-3? (b) How long would it take the sample to decay to 20% of its original amount?

Scientists can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74% as much C radioactivity as does plant material on Earth today. Estimate the age of the parchment.

A curve passes through the point and has the property that the slope of the curve at every point is twice the -coordinate of . What is the equation of the curve?

A roast turkey is taken from an oven when its temperature has reached and is placed on a table in a room where the temperature is . (a) If the temperature of the turkey is after half an hour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to ?

A thermometer is taken from a room where the temperature is C to the outdoors, where the temperature is . After one minute the thermometer reads C. (a) What will the reading on the thermometer be after one more minute? (b) When will the thermometer read C?

When a cold drink is taken from a refrigerator, its temperature is C. After 25 minutes in a C room its temperature has increased to C. (a) What is the temperature of the drink after 50 minutes? (b) When will its temperature be C?

A freshly brewed cup of coffee has temperature C in a C room. When its temperature is C, it is cooling at a rate of C per minute. When does this occur?

The rate of change of atmospheric pressure with respect to altitude is proportional to , provided that the temperature is constant. At C the pressure is kPa at sea level and kPa at m. (a) What is the pressure at an altitude of 3000 m? (b) What is the pressure at the top of Mount McKinley, at an altitude of 6187 m?

(a) If $500 is borrowed at 14% interest, find the amounts due at the end of 2 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) daily, (v) hourly, and (vi) continuously. (b) Suppose $500 is borrowed and the interest is compounded continuously. If is the amount due after years, where , graph for each of the interest rates 14%, 10%, and 6% on a common screen.

If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (a) annually (b) semiannually (c) monthly (d) weekly (e) daily (f ) continuously

(a) How long will it take an investment to double in value if the interest rate is 6% compounded continuously? (b) What is the equivalent annual interest rate?

Prove that .

Simplify the expression. tansin1 x c

Simplify the expression. sintan x tans

Simplify the expression. cscarctan 2x 1

Prove Formula 6 for the derivative of by the same method as for Formula 3.

(a) Prove that . (b) Use part (a) to prove Formula 6.

Prove that .

Prove that .

Prove that .

Find the derivative of the function. Simplify where possible. y stan1 x d

Find the derivative of the function. Simplify where possible.y tan arcsin x 1 sx y

Find the derivative of the function. Simplify where possible.hx s1 x 2 y tan arcsin x 1 sx

Find the derivative of the function. Simplify where possible.y sin fx x lnarctan x 1 19. 2x 1 hx s1 x

Find the derivative of the function. Simplify where possible.fx x lnarctan x 1 19.

Find the derivative of the function. Simplify where possible.Hx 1 x 2 21. arctan x y si

Find the derivative of the function. Simplify where possible.ht esec1t Hx

Find the derivative of the function. Simplify where possible.y cos1 e2x ht

Find the derivative of the function. Simplify where possible.y x cos1 x s1 x 2 y c

Find the derivative of the function. Simplify where possible.y arctancos  y

Find the derivative of the function. Simplify where possible.y tan1 (x s1 x 2 ) y

Find the derivative of the function. Simplify where possible.ht cot1 t cot1 1t y tan1

Find the derivative of the function. Simplify where possible.y tan1  x a  lnx a x a 16

Find the derivative of the function. Simplify where possible.y arccos b a cos x a b cos x , 0 x , a  b  0 y\

Find the derivative of the function. Find the domains of the function and its derivative.f x arcsine 31. 3 2x x y arcc

Find the derivative of the function. Find the domains of the function and its derivative.tx cos1 f x arcsine 31. 3 2x x y arc

Find if .

If , find .

Find an equation of the tangent line to the curve at the point .

Find the limit.lim 2  xl1 sin1 x y

Find the limit.lim x l arccos 1 x 2 1 2x lim 2

Find the limit.lim ln x x l arctanex 37. lim

Find the limit.lim xl0 tan1 lim ln x x l

A ladder 10 ft long leans against a vertical wall. If the bottom of the ladder slides away from the base of the wall at a speed of 2 fts, how fast is the angle between the ladder and the wall changing when the bottom of the ladder is 6 ft from the base of the wall?

A lighthouse is located on a small island, 3 km away from the nearest point on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from ?

Some authors define and . Show that with this definition, we have (instead of the formula given in Exercise 14)

. (a) Sketch the graph of the function . (b) Sketch the graph of the function , (c) Show that . (d) Sketch the graph of , , and find its derivative.

Find the numerical value of each expression. cosh1 sech 0 1 c

Find the numerical value of each expression. sinh1 sinh 1 1 c

Prove the identity. sinhx sinh x sinh1 (This shows that is an odd function.)

Prove the identity. coshx cosh x sinh (This shows that is an even function.)

Prove the identity. cosh x sinh x ex 9. co

Prove the identity. cosh x sinh x ex co

Prove the identity. sinhx y sinh x cosh y cosh x sinh y Find

Prove the identity. coshx y cosh x cosh y sinh x sinh y 16

Prove the identity. sinh 2x 2 sinh x cosh x

Prove the identity. 1 tanh x 1 tanh x e 2x 13

Prove the identity. cosh x sinh x nn cosh nx sinh nx ( any real number) 1

If , find the values of the other hyperbolic functions at .

If , find the values of the other hyperbolic functions at .

a) Use the graphs of , , and in Figures 13 to draw the graphs of , , and . ; (b) Check the graphs that you sketched in part (a) by using a graphing device to produce them.

Prove the formulas given in Table 1 for the derivatives of the functions (a) , (b) , (c) , (d) , and (e) .

Give an alternative solution to Example 3 by letting and then using Exercise 9 and Example 1(a) with replaced by .

Prove Equation 4.

Prove Formula 5 using (a) the method of Example 3 and (b) Exercise 14 with replaced by .

For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Formula 3. (a) (b) (c)

Prove the formulas given in Table 6 for the derivatives of the following functions. (a) (b) (c)

Find the derivative. tx sinh x sech1

Find the derivative. f x x cosh x 2

Find the derivative. Fx sinh x tanh x t

Find the derivative. hx sinhx 2 Fx

Find the derivative. f t et sech t hx

Find the derivative. ht coths1 t 2 f t

Find the derivative. f t lnsinh t ht

Find the derivative. Ht tanhet f t

Find the derivative. y sinhcosh x 35.

Find the derivative. y ecosh 3x y

Find the derivative. y x sx 2 sinh1 2x y

Find the derivative. y tanh1 y x sx 2

Find the derivative. y x tanh1 x ln s1 x 2 y

Find the derivative. y x sinh1 x3 s9 x 2 39. y

Find the derivative. y sech1 s1 x 2 , x  0 y

Find the derivative. y coth1 sx 2 1 y

A flexible cable always hangs in the shape of a catenary , where and are constants and (see Figure 4 and Exercise 44). Graph several members of the family of functions . How does the graph change as varies?

A telephone line hangs between two poles 14 m apart in the shape of the catenary , where and are measured in meters. (a) Find the slope of this curve where it meets the right pole. (b) Find the angle between the line and the pole.

Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve that satisfies the differential equation where is the linear density of the cable, is the acceleration due to gravity, and is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function is a solution of this differential equation.

(a) Show that any function of the form satisfies the differential equation . (b) Find such that , , and .

Evaluate .

At what point of the curve does the tangent have slope 1?

If , show that .

Show that if and , then there exist numbers and such that equals either or . In other words, almost every function of the form is a shifted and stretched hyperbolic sine or cosine function.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 x tan x s

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. im tl0 et 1

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim tl0 e3t 1 t

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 tan px tan qx

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim l2 1 sin  csc  l

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 ln x x

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl ln ln x x

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim tl0 5t 3t t

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl1 ln x sin x

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 e x 1 x x 2 13

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl e x x 3

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl x ln1 2ex lim

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 cos mx cos nx x 2 l

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl1 1 x ln x 1 cos x lim

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 x tan1 4x lim

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl1 x a ax a 1 x 1 2 lim

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 1 e2x sec x li

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 sx ln x

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl x 2 ex li

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim sin x ln x xl0 cot 2x sin 6x

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 lim sin x ln x x

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim x tan1x xl x 3 e x 2

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl lim x tan1x xl

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim 28. csc x cot x xl xe 1x x lim

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 lim 28. csc x cot x xl

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim  xl x ln x l

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl1  1 ln x 1 x 1 l

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 x x 2

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 tan 2x x lim

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 1 2x 1x 33.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl 1 a x  bx

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl0 cos x 1x 2 1

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim xl xln 21 ln x lim

Use a graph to estimate the value of the limit. Then use lHospitals Rule to find the exact value. lim xl x lnx 5 ln x lim

Use a graph to estimate the value of the limit. Then use lHospitals Rule to find the exact value. lim xl4 tan x tan 2x li

Prove that for any positive integer . This shows that the exponential function approaches infinity faster than any power of .

Prove that for any number . This shows that the logarithmic function approaches more slowly than any power of .

If an initial amount of money is invested at an interest rate compounded times a year, the value of the investment after years is If we let , we refer to the continuous compounding of interest. Use lHospitals Rule to show that if interest is compounded continuously, then the amount after years is

If an object with mass is dropped from rest, one model for its speed after seconds, taking air resistance into account, is where is the acceleration due to gravity and is a positive constant. (a) Calculate . What is the meaning of this limit? (b) For fixed , use lHospitals Rule to calculate . What can you conclude about the velocity of a falling object in a vacuum?

If an electrostatic field acts on a liquid or a gaseous polar f  f2 0 f 2 7 dielectric, the net dipole moment per unit volume is Show that . 44. A

A metal cable has radius and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is . The velocity of an electrical impulse in the cable is where is a positive constant. Find the following limits and interpret your answers.

The first appearance in print of lHospitals Rule was in the book Analyse des Infiniment Petits published by the Marquis de lHospital in 1696. This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function as approaches , where . (At that time it was common to write instead of .) Solve this problem.

The figure shows a sector of a circle with central angle . Let be the area of the segment between the chord and the arc . Let be the area of the triangle . Find .

. If is continuous, , and , evaluate lim xl0 f 2 3x f2 5x x 43. If

For what values of and is the following equation true? lim xl0  sin 2x x3 a b x2  0 a

If is continuous, use lHospitals Rule to show that lim hl0 fx h f x h 2h f x 49. f Explain the meaning of this equation with the aid of a diagram.

If is continuous, show that lim h l 0 f x h 2fx f x h h2 f x f  lim hl

Let f x  e1x 2 0 if x 0 if x 0 lim (a) Use the definition of derivative to compute . (b) Show that has derivatives of all orders that are defined on . [Hint: First show by induction that there is a polynomial and a nonnegative integer such that for .]

Let (a) Show that is continuous at . (b) Investigate graphically whether is differentiable at by zooming in several times toward the point on the graph of . (c) Show that is not differentiable at . How can you reconcile this fact with the appearance of the graphs in part (b)?

(a) What is a one-to-one function? How can you tell if a x function is one-to-one by looking at its graph? (b) If is a one-to-one function, how is its inverse function defined? How do you obtain the graph of from the graph of ? (c) Suppose is a one-to-one function. If , write a formula for .

(a) Express as a limit. (b) What is the value of correct to five decimal places? (c) Why is the natural exponential function used more often in calculus than the other exponential functions ? (d) Why is the natural logarithmic function used more often in calculus than the other logarithmic functions ?

(a) What are the domain and range of the natural exponential function ? (b) What are the domain and range of the natural logarithmic function ? (c) How are the graphs of these functions related? Sketch these graphs, by hand, using the same axes. (d) If a is a positive number, , write an equation that expresses in terms of .

(a) How is the inverse sine function defined? What are its domain and range? (b) How is the inverse cosine function defined? What are its domain and range? (c) How is the inverse tangent function defined? What are its domain and range? Sketch its graph.

Write the definitions of the hyperbolic functions , , and .

. State the derivative of each function. (a) (b) (c) (d) (e) (f) (g) (h) (i) ( j) (k) (l) (m)

(a) Write a differential equation that expresses the law of natural growth. What does it say in terms of relative growth rate? (b) Under what circumstances is this an appropriate model for population growth? (c) What are the solutions of this equation?

(a) What does lHospitals Rule say? (b) How can you use lHospitals Rule if you have a product where and as ? (c) How can you use lHospitals Rule if you have a difference where and as ? (d) How can you use lHospitals Rule if you have a power fx where and as ? fx

If , then .

d dx 10x x10x1 Dete

d dx ln 10 1 10 d

The inverse function of is .

cos1 x 1 cos x y

tan1 x sin1 x cos1 x cos

cosh x  1 x

lim xl tan x 1 cos x lim xl sec2 x sin x cos

Solve the equation for . ln x 1 ln x 1 1 d ln1

Solve the equation for . log5c x l = d

Solve the equation for . tan -x = 1

Solve the equation for . sin x = 0.3

Differentiate.f t t 2 ln t ta

Differentiate.t et 1 et f t

Differentiate.h etan 2 tt

Differentiate.hu 10su h

Differentiate.y ln 2 2t 2  sec 5x tan 5x  hu

Differentiate.y et t y ln 2 2t 2  sec 5

Differentiate.y e cxc sin x cos x y e

Differentiate.y sin1 ex y

Differentiate.y lnsec 2 x y

Differentiate.y lnx 2 ex y lnsec 2 x

Differentiate.y xe1x y

Differentiate. y xr esx y

Differentiate.y 2 t 2 y

Differentiate.y ecos x cosex y 2 t 2

Differentiate.Hv v tan 1 v y e

Differentiate.Fz log101 z 2 Hv v

Differentiate.y x sinhx 2 Fz

Differentiate.y cos x x y

Differentiate.y ln sin x y arctan(arcsin sx ) 1 2 sin2 x y

Differentiate.y arctan(arcsin sx ) 1

Differentiate.y ln y y 1 1 x  1 ln x y

Differentiate.xe y ln y y 1 1

Differentiate.y lncosh 3x xe

Differentiate.y x 2 1 4 2x 1 3 3x 1 5 y lncosh

Differentiate.y cosh sx 1 sinh x y

Differentiate.y x tanh1 y cosh sx 1

Differentiate.fx esin3 lnx 2 1 y x ta

Show that d dx 1 2 tan1 x 1 4 ln x 1 2 x 2 1  1 1 x1 x 2 fx esin3 l

Find in terms of .f x e tx f t

Find in terms of .f x tex f x

Find in terms of .f x ln f x tln x  tx f x tex

Find in terms of .f x tln x  tx

Find .f x 2 f x

Find .f x ln2x x f

Use mathematical induction to show that if , then .

Find if .

Find an equation of the tangent to the curve at the given point. y 2 xe 0, 2 y x

Find an equation of the tangent to the curve at the given point.y x ln x e, e x

At what point on the curve is the tangent horizontal?

If , find . Graph and on the same screen and comment.

(a) Find an equation of the tangent to the curve that is parallel to the line . (b) Find an equation of the tangent to the curve that passes through the origin.

The function , where a, b, and K are positive constants and , is used to model the concentration at time t of a drug injected into the bloodstream. (a) Show that . (b) Find , the rate at which the drug is cleared from circulation. (c) When is this rate equal to 0?

A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000?

Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?

Let be the concentration of a drug in the bloodstream. As the body eliminates the drug, decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus , where is a positive number called the elimination constant of the drug. (a) If is the concentration at time , find the concentration at time .t (b) If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug?

A cup of hot chocolate has temperature in a room kept at . After half an hour the hot chocolate cools to . (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to ?

Evaluate the limit.lim xl e3x 4

Evaluate the limit.im xl10 ln100 x 2 lim xl e

Evaluate the limit.lim 3 x xl3 e2x3 lim

Evaluate the limit.lim x l arctanx lim 3 x xl3

Evaluate the limit.lim sin x xl0 lnsinh x lim

Evaluate the limit.lim x l ex lim sin x x

Evaluate the limit.lim x l 1 2x 1 2x l

Evaluate the limit.lim x l 1 4 x

Evaluate the limit.lim xl0 tan x ln1 x li

Evaluate the limit.lim xl0 1 cos x x 2 x li

Evaluate the limit.lim xl0 e4x 1 4x x 2 l

Evaluate the limit.lim xl e4x 1 4x x 2 li

Evaluate the limit.lim ln x xl x3 ex

Evaluate the limit.lim xl0 x 2 lim ln x x

Evaluate the limit.lim x l 1  x x 1 1 ln x

Evaluate the limit.lim x l2 tan x cos x lim x

If , find .

Show that cos arctan sinarccot x x 2 1 x 2 2 f