Solution: ?Explain why or why not Determine whether the following statements are true

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\left.\frac{d}{d x}\left(x \sec ^{-1} x\right)\right|_{x=\frac{2}{\sqrt{3}}}\)

Problem 36RE Implicit differentiation ?Calculate y??(?x?) ?for the following relations. y y= e 1+sin x

37-39. Implicit differentiation Calculate y’(x) for the following relations. \(y=\frac{e^{y}}{1+\sin x}\)

37-39. Implicit differentiation Calculate y’(x) for the following relations. \(\sin (x) \cos (y-1)=\frac{1}{2}\)

Problem 40RE Tangent lines? ?Find an equation of the line tangent to the following curves at the given point. y=3x + sin x ;x = 0

Problem 39 RE Quadratic functions 2 a.? Show th ? a?t i?? ,? ? )) is any point on the gr?ap? h of ?f(x? ?) = x , then the slope of the tangent line at that p?oint i?s m ? = 2?a. b.? Sho? ? t?hat if (?a, ?f(?a)) is any point on t? e? graph ? f ?? ?) =? x + c ? x + ? , then the slope of the tangent line at t?hat poi? nt i?s ?m = 2?ab + c ? .

41-44. Tangent lines Find an equation of the line tangent to the following curves at the given point. \(y=\frac{4x}{x^2+3};\ \ x=3\)

41-44. Tangent lines Find an equation of the line tangent to the following curves at the given point. \(y=3x^3+\sin x;\ \ x=0\)

41-44. Tangent lines Find an equation of the line tangent to the following curves at the given point. \(y+\sqrt{xy}=6;\ \ (x,\ y)=(1,\ 4)\)

Problem 44RE Horizontal tangent line? For what value(s?? is the line tangent to the curve y = x 6?x horizontal?

Problem 45RE 2 A parabola property? ? L? et f? (x? ) = x a.? Show that ? fo?r all? ? y? . b.? Is this prope?rty? true fo?r f? ) = ? ?2 , where a ? is a nonzero real number? c.? Give a geometrical interpretation of this properly. d.? Is this pr?o? erty tru? e for f? (?x? x ?

Problem 46RE Higher-order derivatives? ?? in?d y?, ?y?,? ?and y? ?for the following functions.

47-48. Higher-order derivatives Find y’, y’’, and y’’’ for the following functions. \(y=\sin \sqrt{x}\)

Problem 48RE Derivative formulas? ?Evaluate the following derivatives. Express your answers in terms of f, g, f??? nd g?. d 2 dx(x f(x))

49-52. Derivative formulas Evaluate the following derivatives. Express your answers in terms of f, g, f’ and g’. \(\frac{d}{d x}\left[x^{2} f(x)\right]\)

49-52. Derivative formulas Evaluate the following derivatives. Express your answers in terms of f, g, f’ and g’. \(\frac{d}{d x} \sqrt{\frac{f(x)}{g(x)}}\)

49-52. Derivative formulas Evaluate the following derivatives. Express your answers in terms of f, g, f’ and g’. \(\frac{d}{d x}\left(\frac{x \cdot f(x)}{g(x)}\right)\)

Problem 52 RE Finding derivatives from a table? Find the values of the following derivatives using the table. a.? d [f(X)+2g(x)] b .?d [x f(x] dx x=3 dx g(x) x=1 c.? d[f(g(x )] x=3 dx

Problem 53RE Limits? ?The following limits represent the derivative of a function f at a point a. Find a possible f and a, and then evaluate the limit. 2 ? 1 lim sin (4+h)?2 h?0 h

Problem 55RE Derivative of the inverse at a point? ?Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point. 1 f? x)=x+1 a t? (0)

54-55. Limits The following limits represent the derivative of a function f at a point a. Find a possible f and a, and then evaluate the limit. \(\lim _{h \rightarrow 0} \frac{\sin ^{2}\left(\frac{\pi}{4}+h\right)-\frac{1}{2}}{h}\)

56–57. Derivative of the inverse at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point. f(x) = 1/(x + 1) at f(0)

Problem 57 RE Derivative of the inverse? ?Find the derivative of the inverse of the following functions. Express the result with x as the independent variable. f?(?x?) = 12?x? ? 16

58-59. Derivative of the inverse Find the derivative of the inverse of the following functions. Express the result with x as the independent variable. f(x) = 12x - 16

Problem 59RE A function and its inverse function? The function f(x)= x+1 is one-to-one for x? > ? 1 and has an inverse on that interval. a.? Graph ?f? for ?x? > ?1. ?1 b.? Find the inverse function f corresponding to the function graphed in part (a). Graph f ?1 on the same set of axes as in part (a). ?1 c.? Evaluate the derivative of f ? t the point . ?1 d.? Sketch the tangent lines on the graphs of ?f? and f at and , respectively.

Problem 61 RE Velocity of a rocket? The height in feet of a rocket above the ground is given by s(t)= 202t2 for t? 0. t +1 a.? Graph the height function and describe the motion of the rocket. b.? Find the velocity of t? h?e roc ? k? t? (?t) =? ?(?t). c.? Graph the velocity function and determine the approximate time at which the velocity is a maximum.

Problem 62 RE Marginal and average cost Suppose? the cost of producing ?x? lawnmowers is C?(? ?) = ?0.02x + 400?x? + 5000. a.? Determine the average and marginal costs for ?x? = 3000 lawnmowers. b.? Interpret the meaning of your results in part (a).

Problem 63 RE 3 2 Population growth? Suppose ?p?(?t?) = ?1.7t + 72t + 7200t ? ? + 80,000 is the population of a City t? ?years after 1950. a.? Determine the average rate of growth of the city from 1950 to 2000. b.? What was the rate of growth of the city in 1990?

Problem 64 RE Position of a piston? The distance between the head of a piston and the end of a 8t cylindrical chamber is given by ?x(t)= t+1c? ?, for ?t ? 0 s. The radius of the cylinder is 4 cm. a.? Find the volume of the cha ? mber for ? ? 0. b.? Find the rate of change of t? he? volu?me ?V?(?t) for ?t ? 0. c.? Graph the derivative of the volume function. On what intervals is the volume increasing? Decreasing?

Problem 65 RE Boat rates? Two boats leave a dock at the same lime. One boat travels south at 30 mi/hr and the other travels east at 40 mi/hr. After half an hour, how fast is the the distance between the boats increasing?

Problem 66 RE Rate of descent of a hot-air balloon? A rope is attached to the bottom of a hot-air balloon that is floating above a flat field. If the angle of the rope to the ground remains at 65° and the rope is pulled in at 5 ft/s. how quickly is the elevation of the balloon changing?

Problem 67 RE 3 Filling a tank? Water flows into a conical tank at a rate of 2 ft /min. If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.

Problem 68RE Angle of elevation? A jet flies horizontally 500 ft directly above a spectator at an air show at 450 mi/hr. Determine how quickly the angle of elevation (between the ground and the line from the spectator to the jet) is changing 2 s later.

Problem 69 RE Viewing angle? A man whose eye level is 6 ft above the ground walks toward a billboard at a rate of 2 ft/s. The bottom of the billboard is 10 ft above the ground and it is 15 ft high. The man’s viewing angle is the angle formed by the lines between the man’s eyes and the top and bottom of the billboard. At what rate is the viewing angle changing when the man is 30 ft from the billboard?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function f(x) = |2x + 1| is continuous for all x; therefore, it is differentiable for all x. b. If \(\frac{d}{d x}[f(x)]=\frac{d}{d x}[g(x)]\), then f = g. c. For any function \(f,\ \frac{d}{dx}|f(x)|=\left|f^{\prime}(x)\right|\). d. The value of f’(a) fails to exist only if the curve y = f(x) has a vertical tangent line at x = a. e. An object can have negative acceleration and increasing speed.

2-5 Tangent lines a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P. b. Find an equation of the line tangent to the curve y = f(x) at point P; then, graph the curve and the tangent line. \(f(x)=4x^2-7x+5;\ \ P(2,\ 7)\)

2-5 Tangent lines a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P. b. Find an equation of the line tangent to the curve y = f(x) at point P; then, graph the curve and the tangent line. \(f(x)=5x^3+x;\quad\ P(1,\ 6)\)

2-5 Tangent lines a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P. b. Find an equation of the line tangent to the curve y = f(x) at point P; then, graph the curve and the tangent line. \(y=f(x)=\frac{x+3}{2x+1};\ \ P(0,\ 3)\)

2-5 Tangent lines a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P. b. Find an equation of the line tangent to the curve y = f(x) at point P; then, graph the curve and the tangent line. \(f(x)=\frac{1}{2\sqrt{3x+1}};\ P\left(0,\ \frac{1}{2}\right)\)

Calculating average and instantaneous velocities Suppose the heights of an object (in m) above the ground after t seconds is approximated by the function \(s=f(t)=-4.9 t^{2}+25 t+1\). a. Make a table showing the average velocities of the object from time t = 1 to t = 1 + h for h = 0.01, 0.001, 0.0001, and 0.00001. b. Use the table in part (a) to estimate the instantaneous velocity of the object at t = 1 s. c. Use limits to verify your estimate in part (b).

Population of the United States in the 20th century The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1900. These data are plotted and connected with a smooth curve y = p(t) in the figure. a. Compute the average rate of population growth from 1950 to 1960. b. Explain why the average rate of growth from 1950 to 1960 is a good approximation to the instantaneous) rate of growth in 1955. c. Estimate the instantaneous rate of growth in 1985.

Growth rate of bacteria Suppose the following graph represents the number of bacteria in a culture t hours after the start of an experiment. a. At approximately what time is the instantaneous growth rate the greatest? Estimate the growth rate at this time. b. At approximately what time in the interval \(0 \leq t \leq 36\) is the instantaneous growth rate the least? Estimate the instantaneous growth rate at this time. c. What is the average growth rate over the interval \(0 \leq t \leq 36\)?

Velocity of a skydiver Assume the graph on the next page represents the distance (in m) fallen by a skydiver t seconds after jumping out of a plane. a. Estimate the velocity of the skydiver at t = 15 s. b. Estimate the velocity of the skydiver at t = 70 s. c. Estimate the average velocity of the skydiver between t = 20 s and t = 90 s. d. Sketch a graph of the velocity function for \(0 \leq t \leq 120\). e. What significant event do you think occurred at t = 30?

10-11. Using the definition of the derivative Use the limit definition of the derivative to do the following. Verify that \(f^{\prime}(x)=4 x-3\) if \(f(x)=2 x^{2}-3 x+1\)

10-11. Using the definition of the derivative Use the limit definition of the derivative to do the following. Verify that \(g^{\prime}(x)=\frac{1}{\sqrt{2 x-3}}\) if \(g(x)=\sqrt{2 x-3}\)

Sketching a derivative graph Sketch a graph of f' for the function f shown in the figure.

Sketching a derivative graph Sketch a graph of g' for the function g shown in the figure.

Matching functions and derivatives Match the functions in (a)-(d) with the derivatives (A)-(D).

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(\frac{2}{3} x^{3}+\pi x^{2}+7 x+1\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(2 x \sqrt{x^{2}-2 x+2}\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d t}\left(5 t^{2} \sin t\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(5 x+\sin ^{3} x+\sin x^{3}\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d \theta}\left[4 \tan \left(\theta^{2}+3 \theta+2\right)\right]\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(\csc ^{5} 3 x\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d u}\left(\frac{4 u^{2}+u}{8 u+1}\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d t}\left(\frac{3 t^{2}-1}{3 t^{2}+1}\right)^{-3}\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d \theta}[\tan (\sin \theta)]\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d v}\left(\frac{v}{3 v^{2}+2 v+1}\right)^{1 / 3}\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}[2 x(\sin x) \sqrt{3 x-1}]\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(x e^{-10 x}\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(x \ln ^{2} x\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d w}\left(e^{-w} \ln w\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(2^{x^{2}-x}\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(\log _{3}(x+8)\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left[\sin ^{-1}\left(\frac{1}{x}\right)\right]\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(\frac{d}{d x}\left(x^{\sin x}\right)\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(f^{\prime}(1)\) when \(f(x)=x^{1 / x}\)

15-36. Evaluating derivatives Evaluate and simplify the following derivatives. \(f^{\prime}(1)\) when \(f(x)=\tan ^{-1}\left(4 x^{2}\right)\)

A function and its inverse function The function \(f(x)=\frac{x}{x+1}\) is one-to-one for x > -1 and has an inverse on that interval. a. Graph f for x > -1. b. Find the inverse function \(f^{-1}\) corresponding to the function graphed in part (a). Graph \(f^{-1}\) on the same set of axes as in part (a). c. Evaluate the derivative of \(f^{-1}\) at the point \(\left(\frac{1}{2},\ 1\right)\). d. Sketch the tangent lines on the graphs of f and \(f^{-1}\) at \(\left(1,\ \frac{1}{2}\right)\) and \(\left(\frac{1}{2},\ 1\right)\), respectively.

Filling a tank Water flows into a conical tank at a rate of \(2\ \mathrm{ft}^3/\mathrm{\min}\). If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.

Angle of elevation A jet flies horizontally 500 ft directly above a spectator at an air show at 450 mi/hr. Determine how quickly the angle of elevation (between the ground and the line from the spectator to the jet) is changing 2 s later.

Viewing angle A man whose eye level is 6 ft above the ground walks toward a billboard at a rate of 2 ft/s. The bottom of the billboard is 10 ft above the ground and it is 15 ft high. The man's viewing angle is the angle formed by the lines between the man's eyes and the top and bottom of the billboard. At what rate is the viewing angle changing when the man is 30 ft from the billboard?