?If y = f (x) and x changes from x1 to x2, write expressions for the following.(a) The

Explain what each of the following means and illustrate with a sketch. 1. 2. 3. 4. 5.

Describe several ways in which a limit can fail to exist. Illustrate with sketches.

State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f ) Power Law (g) Root Law

What does the Squeeze Theorem say?

(a) What does it mean to say that the line x = a is a vertical asymptote of the curve y = f(x)? Draw curves to illustrate the various possibilities. (b) What does it mean to say that the line y = L is a horizontal asymptote of the curve y = f(x)? Draw curves to illustrate the various possibilities.

Which of the following curves have vertical asymptotes? Which have horizontal asymptotes? 1. y = x4 2. y = sin x 3. y = tan x 4. y = tan-1x 5. y = ex 6. y = ln x 7. y = 1/x 8. y =

(a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval (-?, ?)? What can you say about the graph of such a function?

(a) Give examples of functions that are continuous on [-1, 1]. (b) Give an example of a function that is not continuous on [0, 1].

What does the Intermediate Value Theorem say?

Write an expression for the slope of the tangent line to the curve y = f(x) at the point (a, f (a)).

Suppose an object moves along a straight line with position f (t) at time t. Write an expression for the instantaneous velocity of the object at time t = a. How can you interpret this velocity in terms of the graph of f ?

If y = f (x) and x changes from x1 to x2, write expressions for the following. (a) The average rate of change of y with respect to x over the interval [x1, x2] (b) The instantaneous rate of change of y with respect to x at x = x1

Define the derivative f’(a). Discuss two ways of interpreting this number.

Define the second derivative of f. If f(t) is the position function of a particle, how can you interpret the second derivative?

(a) What does it mean for f to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? (c) Sketch the graph of a function that is continuous but not differentiable at a = 2.

Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [-1, 1] and f(-1) = 4 and f (1) = 3, then there exists a number r such that |r | < 1 and f (r) = ???? .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Let f be a function such that f(x) = 6. Then there exists a positive number ???? such that if 0 < | x | < ????, then | f(x) - 6 | < 1.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(x) > 1 for all x and f(x) exists, then f(x) > 1.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous at a, then f is differentiable at a.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f’(r) exists, then f(x) = f(r).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The equation \(x^{10}-10 x^{2}+5=0\) has a solution in the interval (0,2).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous at a, so is | f |.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If | f | is continuous at a, so is f.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is differentiable at a, so is | f |.

Prove the statement using the precise definition of a limit. \(\lim _{x \rightarrow 2}\left(x^{2}-3 x\right)=-2\) Text Transcription: lim _{x rightarrow 2} (x^2 - 3x) = -2

Prove the statement using the precise definition of a limit.

Let (a) Evaluate each limit, if it exists. (i) (ii) (iii) (iv) (v) (vi) (b) Where is f discontinuous? (c) Sketch the graph of f.

Let (a) For each of the numbers 2, 3, and 4, determine whether t is continuous from the left, continuous from the right, or continuous at the number. (b) Sketch the graph of t.

Show that the function is continuous on its domain. State the domain. \(h(x)=x e^{\sin x}\)

Show that the function is continuous on its domain. State the domain.

Use the Intermediate Value Theorem to show that there is a solution of the equation in the given interval. \(x^{5}-x^{3}+3 x-5=0\), (1,2)

Use the Intermediate Value Theorem to show that there is a solution of the equation in the given interval. , (0, 1)

(a) Find the slope of the tangent line to the curve \(y=9-2 x^{2}\) at the point (2, 1). (b) Find an equation of this tangent line.

Find equations of the tangent lines to the curve at the points with x-coordinates 0 and -1.

The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + ¼ t2, where t is measured in seconds. (a) Find the average velocity over each time period. (i) [1, 3] (ii) [1, 2] (iii) [1, 1.5] (iv) [1, 1.1] (b) Find the instantaneous velocity when t = 1.

According to Boyle’s Law, if the temperature of a confined gas is held fixed, then the product of the pressure P and the volume V is a constant. Suppose that, for a certain gas, PV = 800, where P is measured in pounds per square inch and V is measured in cubic inches. (a) Find the average rate of change of P as V increases from 200 in3 to 250 in3. (b) Express V as a function of P and show that the instantaneous rate of change of V with respect to P is inversely proportional to the square of P.

(a) Use the definition of a derivative to find f’(2), where f(x) = x3 - 2x. (b) Find an equation of the tangent line to the curve y = x3 - 2x at the point (2, 4). (c) Illustrate part (b) by graphing the curve and the tangent line on the same screen.

Find a function f and a number a such that

The total cost of repaying a student loan at an interest rate of r% per year is C = f(r). (a) What is the meaning of the derivative f’(r)? What are its units? (b) What does the statement f’(10) = 1200 mean? (c) Is f’(r) always positive or does it change sign?

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Find the derivative of f using the definition of a derivative. What is the domain of f’?

Find the derivative of f using the definition of a derivative. What is the domain of f’?

(a) If \(f(x)=\sqrt{3-5 x}\), use the definition of a derivative to find \(f^{\prime}(x)\). (b) Find the domains of f and f’. (c) Graph f and f’ on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable.

(a) Find the asymptotes of the graph of and use them to sketch the graph. (b) Use your graph from part (a) to sketch the graph of f’. (c) Use the definition of a derivative to find f’(x). (d) Graph f’ and compare with your sketch in part (b).

The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.

The figure shows the graphs of f, f’, and f’’. Identify each curve, and explain your choices.

Sketch the graph of a function f that satisfies all of the following conditions: The domain of f is all real numbers except 0, , , f’(x) > 0 for all x in the domain of f, ,

Let P(t) be the percentage of Americans under the age of 18 at time t. The table gives values of this function in census years from 1950 to 2010. (a) What is the meaning of P’(t)? What are its units? (b) Construct a table of estimated values for P’(t). (c) Graph P and P’. (d) How would it be possible to get more accurate values for P’(t)?

Let B(t) be the number of US $20 bills in circulation at time t. The table gives values of this function from 1995 to 2015, as of December 31, in billions. Interpret and estimate the value of \(B \prime\)(2010).

The total fertility rate at time t, denoted by F(t), is an estimate of the average number of children born to each woman (assuming that current birth rates remain constant). The graph of the total fertility rate in the United States shows the fluctuations from 1940 to 2010. (a) Estimate the values of F’(1950), F’(1965), and F’(1987). (b) What are the meanings of these derivatives? (c) Can you suggest reasons for the values of these derivatives?

Suppose that | f(x) | ? g(x) for all x, where g(x) = 0. Find f(x).

Let f(x) = [[ x ]] + [[ -x ]]. (a) For what values of a does f(x) exist? (b) At what numbers is f discontinuous?

Explain what each of the following means and illustrate with a sketch. (a) \(\lim \limits_{x \rightarrow a} f(x)=L\) (b) \(\lim \limits_{x \rightarrow a^{+}} f(x)=L\) (c) \(\lim \limits_{x \rightarrow a^{-}} f(x)=L\) (d) \(\lim \limits_{x \rightarrow a} f(x)=\infty\) (e) \(\lim \limits_{x \rightarrow \infty} f(x)=L\)

Describe several ways in which a limit can fail to exist. Illustrate with sketches.

State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f ) Power Law (g) Root Law

What does the Squeeze Theorem say?

(a) What does it mean to say that the line x = a is a vertical asymptote of the curve y = f(x)? Draw curves to illustrate the various possibilities. (b) What does it mean to say that the line y = L is a horizontal asymptote of the curve y = f(x)? Draw curves to illustrate the various possibilities.

Which of the following curves have vertical asymptotes? Which have horizontal asymptotes? 1. y = x4 2. y = sin x 3. y = tan x 4. y = tan-1x 5. y = ex 6. y = ln x 7. y = 1/x 8. y =

(a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval (-?, ?)? What can you say about the graph of such a function?

(a) Give examples of functions that are continuous on [-1, 1]. (b) Give an example of a function that is not continuous on [0, 1].

What does the Intermediate Value Theorem say?

Write an expression for the slope of the tangent line to the curve y = f(x) at the point (a, f (a)).

Suppose an object moves along a straight line with position f(t) at time t. Write an expression for the instantaneous velocity of the object at time t = a. How can you interpret this velocity in terms of the graph of f?

If y = f (x) and x changes from x1 to x2, write expressions for the following. (a) The average rate of change of y with respect to x over the interval [x1, x2] (b) The instantaneous rate of change of y with respect to x at x = x1

Define the derivative \(f^{\prime}(a)\). Discuss two ways of interpreting this number.

Define the second derivative of f. If f(t) is the position function of a particle, how can you interpret the second derivative?

(a) What does it mean for f to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? (c) Sketch the graph of a function that is continuous but not differentiable at a = 2.

Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\frac{x^{2}-9}{x-3}=x+3\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(x) = 2 and g(x) = 0, then [ f(x)/g(x) ] does not exist.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(x) = 0 and g(x) = 0, then [ f(x)/g(x) ] does not exist.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If neither f(x) nor g(x) exists, then [ f(x) + g(x) ] does not exist.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(x) exists but g(x) does not exist, then [ f(x) + g(x) ] does not exist.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If p is a polynomial, then p(x) = p(b).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(x) = ? and g(x) = ?, then [ f(x) - g(x) ] = 0.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A function can have two different horizontal asymptotes.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f has domain [0, ?) and has no horizontal asymptote, then f(x) = ? or f(x) = -?.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If the line x = 1 is a vertical asymptote of y = f (x), then f is not defined at 1.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f (1) > 0 and f (3) < 0, then there exists a number c between 1 and 3 such that f (c) = 0.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous at 5 and f (5) = 2 and f (4) = 3, then f(4x2 - 11) = 2.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [-1,1] and f(-1)=4 and f(1)=3, then there exists a number r such that |r|<1 and \(f(r)=\pi\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Let f be a function such that f(x) = 6. Then there exists a positive number ???? such that if 0 < | x | < ????, then | f(x) - 6 | < 1.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(x) > 1 for all x and f(x) exists, then f(x) > 1.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous at a, then f is differentiable at a.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f^{\prime}(r)\) exists, then \(\lim _{x \rightarrow r} f(x)=f(r)\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The equation x10 - 10x2 + 5 = 0 has a solution in the interval (0, 2).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous at a, so is | f |.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If | f | is continuous at a, so is f.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is differentiable at a, so is | f |.

The graph of f is given. (a) Find each limit, or explain why it does not exist. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (b) State the equations of the horizontal asymptotes. (c) State the equations of the vertical asymptotes. (d) At what numbers is f discontinuous? Explain.

Sketch the graph of a function f that satisfies all of the following conditions: , , , , , f is continuous from the right at 3

Find the limit.

Find the limit.

Find the limit.

Find the limit.

Find the limit. \(\lim \limits_{h \rightarrow 0} \frac{(h-1)^{3}+1}{h}\)

Find the limit.

Find the limit.

Find the limit.

Find the limit.

Find the limit.

Find the limit. \(\lim \limits_{x \rightarrow \infty} \frac{\sqrt{x^{2}-9}}{2 x-6}\)

Find the limit. \(\lim _{x \rightarrow-\infty} \frac{\sqrt{x^2-9}}{2 x-6}\)

Find the limit. \(\lim \limits_{x \rightarrow \pi^{-}} \ln (\sin x)\)

Find the limit.

Find the limit.

Find the limit. \(\lim _{x \rightarrow \infty} e^{x-x^2}\)

Find the limit.

Find the limit.

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

If 2x - 1 ? f(x) ? x2 for 0 < x < 3, find .

Prove that \(\lim _{x \rightarrow 0} x^2 \cos \left(1 / x^2\right)=0\).

Prove the statement using the precise definition of a limit.

Prove the statement using the precise definition of a limit. \(\lim \limits_{x \rightarrow 0} \sqrt[3]{x}=0\)

Prove the statement using the precise definition of a limit.

Prove the statement using the precise definition of a limit. \(\lim _{x \rightarrow 4^{+}} \frac{2}{\sqrt{x-4}}=\infty\)

Let (a) Evaluate each limit, if it exists. (i) (ii) (iii) (iv) (v) (vi) (b) Where is f discontinuous? (c) Sketch the graph of f.

Let \(g(x)= \begin{cases}2 x-x^2 & \text { if } 0 \leqslant x \leqslant 2 \\ 2-x & \text { if } 2<x \leqslant 3 \\ x-4 & \text { if } 3<x<4 \\ \pi & \text { if } x \geqslant 4\end{cases}\) (a) For each of the numbers 2, 3, and 4, determine whether g is continuous from the left, continuous from the right, or continuous at the number. (b) Sketch the graph of g.

Show that the function is continuous on its domain. State the domain. h(x) = xesin x

Show that the function is continuous on its domain. State the domain.

Use the Intermediate Value Theorem to show that there is a solution of the equation in the given interval. X5 - x3 + 3x - 5 = 0, (1,2)

Use the Intermediate Value Theorem to show that there is a solution of the equation in the given interval. \(\cos \sqrt{x}=e^{x}-2\), (0,1)

(a) Find the slope of the tangent line to the curve y = 9 - 2x2 at the point (2, 1). (b) Find an equation of this tangent line.

Find equations of the tangent lines to the curve at the points with x-coordinates 0 and -1.

The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + ¼ t2, where t is measured in seconds. (a) Find the average velocity over each time period. (i) [1, 3] (ii) [1, 2] (iii) [1, 1.5] (iv) [1, 1.1] (b) Find the instantaneous velocity when t = 1.

According to Boyle’s Law, if the temperature of a confined gas is held fixed, then the product of the pressure P and the volume V is a constant. Suppose that, for a certain gas, PV = 800, where P is measured in pounds per square inch and V is measured in cubic inches. (a) Find the average rate of change of P as V increases from 200 in3 to 250 in3. (b) Express V as a function of P and show that the instantaneous rate of change of V with respect to P is inversely proportional to the square of P.

(a) Use the definition of a derivative to find f’(2), where f(x) = x3 - 2x. (b) Find an equation of the tangent line to the curve y = x3 - 2x at the point (2, 4). (c) Illustrate part (b) by graphing the curve and the tangent line on the same screen.

Find a function f and a number a such that

The total cost of repaying a student loan at an interest rate of r% per year is C = f(r). (a) What is the meaning of the derivative f’(r)? What are its units? (b) What does the statement f’(10) = 1200 mean? (c) Is f’(r) always positive or does it change sign?

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Find the derivative of f using the definition of a derivative. What is the domain of f’?

Find the derivative of f using the definition of a derivative. What is the domain of \(\mathrm{f}^{\prime}\) ? \(f(t)=\frac{1}{\sqrt{t+1}}\)

(a) If f(x) = , use the definition of a derivative to find f’(x). (b) Find the domains of f and f’. (c) Graph f and f’ on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable.

(a) Find the asymptotes of the graph of and use them to sketch the graph. (b) Use your graph from part (a) to sketch the graph of f’. (c) Use the definition of a derivative to find f’(x). (d) Graph f’ and compare with your sketch in part (b).

The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.

Evaluate .

Find numbers a and b such that .

Evaluate .

The figure shows a point P on the parabola \(y=x^{2}\) and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.

Evaluate the following limits, if they exist, where [[x]] denotes the greatest integer function. 1. (b) x

Sketch the region in the plane defined by each of the following equations. (a) \([[x]]^{2}+[[y]]^{2}=1\) (b) \([[x]]^{2}-[[y]]^{2}=3\) (c) \([[x+y]]^{2}=1\) (d) \([[x]]+[[y]]=1\)

Let \(f(x)-x /[[x]]\) (a) Find the domain and range of \(f\). (b) Evaluate \(\lim _{x \rightarrow \infty} \cdot f(x)\)

A fixed point of a function f is a number c in its domain such that f(c) = c. (The function doesn’t move c; it stays fixed.) 1. Sketch the graph of a continuous function with domain [0,1] whose range also lies in [0,1]. Locate a fixed point of f. 2. Try to draw the graph of a continuous function with domain [0,1] and range in [0,1] that does not have a fixed point. What is the obstacle? 3. Use the Intermediate Value Theorem to prove that any continuous function with domain [0,1] and range in [0,1] must have a fixed point.

If limxa [f(x) + g(x)] = 2 and limxa [f(x) - g(x) = 1, find limxa [f(x)g(x)]..

1. The figure shows an isosceles triangle ABC with = . The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fixed but the altitude |AM| of the triangle approaches 0, so A approaches the midpoint M of BC. What happens to P during this process? Does it have a limiting position? If so, find it. 2. Try to sketch the path traced out by P during this process. Then find an equation of this curve and use this equation to sketch the curve.

1. If we start from 0° latitude and proceed in a westerly direction, we can let T(x) denote the temperature at the point x at any given time. Assuming that T is a continuous function of x, show that at any fixed time there are at least two diametrically opposite points on the equator that have exactly the same temperature. 2. Does the result in part (a) hold for points lying on any circle on the earth’s surface? 3. Does the result in part (a) hold for barometric pressure and for altitude above sea level?

If f is a differentiable function and g(x) = x f(x), use the definition of a derivative to show that \(g^{\prime}(x)=x f^{\prime}(x)+f(x)\).

Suppose f is a function that satisfies the equation f(x + y) = f(x) + f(y) + x2y + xy2 for all real numbers x and y. Suppose also that (a) Find f(0). (b) Find f’(0). (c) Find f’(x).

Suppose f is a function with the property that | f(x) | x2 for all x. Show that f(0) = 0. Then show that f’/(0) = 0.