Chapter 2

A curve has equation \( y = f(x) \). (a) Write an expression for the slope of the secant line through the points \( P(3, f(3)) \) and \( Q(x, f(x)) \). (b) Write an expression for the slope of the tangent line at \( P \).

Explain in your own words the meaning of each of the following. (a) \( \displaystyle \lim_{x \to \infty} f(x) = 5 \) (b) \( \displaystyle \lim_{x \to - \infty} f(x) = 3 \)

Write an equation that expresses the fact that a function \( f \) is continuous at the number 4.

Given that \( \displaystyle \lim_{x \to 2}f(x) = 4 \) \( \displaystyle \lim_{x \to 2}g(x) = -2 \) \( \displaystyle \lim_{x \to 2}h(x) = 0 \) find the limits that exist. If the limit does not exist, explain why. (a) \( \displaystyle \lim_{x \to 2}[f(x) + 5g(x)] \) (b) \( \displaystyle \lim_{x \to 2}[g(x)]^3 \) (c) \( \displaystyle \lim_{x \to 2}\sqrt{f(x)} \) (d) \( \displaystyle \lim_{x \to 2}\frac{3f(x)}{g(x)} \) (e) \( \displaystyle \lim_{x \to 2}\frac{g(x)}{h(x)} \) (f) \( \displaystyle \lim_{x \to 2}\frac{g(x)h(x)}{f(x)} \)

Explain in your own words what is meant by the equation \( \displaystyle \lim_{x\to 2} f(x) = 5 \) Is it possible for this statement to be true and yet \( f(2) = 3 \)? Explain.

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume \(V\) of water remaining in the tank (in gallons) after \(t\) minutes. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline t(\mathrm{~min}) & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline V(\mathrm{gal}) & 694 & 444 & 250 & 111 & 28 & 0 \\ \hline \end{array} $$ (a) If \(P\) is the point (15,250) on the graph of \(V\), find the slopes of the secant lines \(P Q\) when \(Q\) is the point on the graph with \(t=5,10,20,25,\) and 30 (b) Estimate the slope of the tangent line at \(P\) by averaging the slopes of two secant lines. (c) Use a graph of the function to estimate the slope of the tangent line at \(P\). (This slope represents the rate at which the water is flowing from the tank after 15 minutes.)

Graph the curve \( y = e^x \) in the viewing rectangles \( [-1, 1] \) by \( [0, 2] \), \( [-0.5, 0.5] \) by \( [0.5, 1.5] \), and \( [-0.1, 0.1] \) by \( [0.9, 1.1] \). What do you notice about the curve as you zoom in toward the point \( (0, 1) \)?

(a) Can the graph of \(y=f(x)\) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. (b) How many horizontal asymptotes can the graph of \(y=f(x)\) have? Sketch graphs to illustrate the possibilities.

If \( f \) is continuous on \( (-\infty, \infty) \), what can you say about its graph?

Explain what it means to say that \( \displaystyle \lim_{x\to1^-}f(x) = 3 \) and \( \displaystyle \lim_{x\to1^+}f(x) = 7 \) In this situation is it possible that \( \displaystyle \lim_{x\to1}f(x) \) exists? Explain.

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after \( t \) minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. $$ \begin{array}{|l|c|c|c|c|c|} \hline t \text { (min) } & 36 & 38 & 40 & 42 & 44 \\ \hline \text { Heartbeats } & 2530 & 2661 & 2806 & 2948 & 3080 \\ \hline \end{array} $$ The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of \( t \). (a) \( t = 36 \) and \( t = 42 \) (b) \( t = 38 \) and \( t = 42 \) (c) \( t = 40 \) and \( t = 42 \) (d) \( t = 42 \) and \( t = 44 \) What are your conclusions?

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{x \to 3}(5x^3 - 3x^2 + x - 6) \)

Explain the meaning of each of the following. (a) \( \displaystyle \lim_{x\to-3}f(x) = \infty \) (b) \( \displaystyle \lim_{x\to4^+}f(x) = - \infty \)

The point \( P(2, -1) \) lies on the curve \( y = 1/(1-x) \). (a) If \( Q \) is the point \( (x, 1/(1-x)) \), use your calculator to find the slope of the secant line \( PQ \) (correct to six decimal places) for the following values of \( x \): (i) \( 1.5 \) (ii) \( 1.9 \) (iii) \( 1.99 \) (iv) \( 1.999 \) (v) \( 2.5 \) (vi) \( 2.1 \) (vii) \( 2.01 \) (viii) \( 2.001 \) (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at \( P(2, -1) \). (c) Using the slope from part (b), find an equation of the tangent line to the curve at \( P(2, -1) \).

(a) Find the slope of the tangent line to the curve \( y = x - x^3 \) at the point \( (1, 0) \) (i) using Definition 1 (ii) using Equation 2 (b) Find an equation of the tangent line in part (a). (c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at \( (1, 0) \) until the curve and the line appear to coincide.

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{x \to -1}(x^4 - 3x)(x^2 + 5x + 3) \)

The point \( P(0.5, 0) \) lies on the curve \( y = \cos \pi x \). (a) If \( Q \) is the point \( (x, \cos \pi x) \), use your calculator to find the slope of the secant line \( PQ \) (correct to six decimal places) for the following values of \( x \): (i) \( 0 \) (ii) \( 0.4 \) (iii) \( 0.49 \) (iv) \( 0.499 \) (v) \( 1 \) (vi) \( 0.6 \) (vii) \( 0. 51 \) (viii) \( 0.501 \) (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at \( P(0.5, 0) \). (c) Using the slope from part (b), find an equation of the tangent line to the curve at \( P(0.5, 0) \). (d) Sketch the curve, two of the secant lines, and the tangent line.

Find an equation of the tangent line to the curve at the given point. \( y = 4x - 3x^2 \), \( (2, -4) \)

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 0} f(x) = -\infty \), \( \displaystyle \lim_{x \to -\infty} f(x) = 5 \), \( \displaystyle \lim_{x \to \infty} f(x) = -5 \)

Sketch the graph of a function \( f \) that is continuous except for the stated discontinuity. Discontinuous, but continuous from the right, at 2.

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{t \to -2}\frac{t^4 - 2}{2t^2 - 3t + 2} \)

If a ball is thrown into the air with a velocity of \( 40 ft/s \), its height in feet \( t \) seconds later is given by \( y = 40t - 16t^2 \). (a) Find the average velocity for the time period beginning when \( t = 2 \) and lasting (i) 0.5 seconds (ii) 0.1 seconds (iii) 0.05 seconds (iv) 0.01 seconds (b) Estimate the instantaneous velocity when \( t = 2 \).

Find an equation of the tangent line to the curve at the given point. \( y = x^3 - 3x + 1 \), \( (2, 3) \)

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 2} f(x) = \infty \), \( \displaystyle \lim_{x \to 2^+} f(x) = \infty \), \( \displaystyle \lim_{x \to 2^-} f(x) = -\infty \), \( \displaystyle \lim_{x \to -\infty} f(x) = 0 \), \( \displaystyle \lim_{x \to \infty} f(x) = 0 \), \( f(0) = 0 \)

Sketch the graph of a function \( f \) that is continuous except for the stated discontinuity. Discontinuities at -1 and 4, but continuous from the left at -1 and from the right of 4

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{u \to -2}\sqrt{u^4 + 3u + 6} \)

If a rock is thrown upward on the planet Mars with a velocity of \( 10 m/s \), its height in meters \( t \) seconds after is given by \( y = 10t - 1.86t^2 \). (a) Find the average velocity over the given time intervals: (i) \( [1, 2] \) (ii) \( [1, 1.5] \) (iii) \( [1, 1.1] \) (iv) \( [1, 1.01] \) (v) \( [1, 1.001] \) (b) Estimate the instantaneous velocity when \( t = 1 \).

Find an equation of the tangent line to the curve at the given point. \( y = \sqrt{x} \), \( (1, 1) \)

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 2} f(x) = -\infty \), \( \displaystyle \lim_{x \to \infty} f(x) = \infty \), \( \displaystyle \lim_{x \to -\infty} f(x) = 0 \), \( \displaystyle \lim_{x \to 0^+} f(x) = \infty \), \( \displaystyle \lim_{x \to 0^-} f(x) = -\infty \),

Sketch the graph of a function \( f \) that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5.

For the limit $$ \lim_{x \to 2}(x^3 - 3x + 4) = 6 $$ illustrate Definition 2 by finding values of \( \delta \) that correspond to \( \varepsilon = 0.2 \) and \( \varepsilon = 0.1 \).

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{x \to 8}(1 + \sqrt[3]{x})(2 - 6x^2 + x^3) \)

Find an equation of the tangent line to the curve at the given point. \( y = \dfrac{2x + 1}{x + 2} \), \( (1, 1) \)

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to \infty} f(x) = 3 \), \( \displaystyle \lim_{x \to 2^-} f(x) = \infty \), \( \displaystyle \lim_{x \to 2^+} f(x) = -\infty \), \( f \) is odd

For the limit $$ \lim_{x \to 0}\frac{e^{2x} - 1}{x} = 2 $$ illustrate Definition 2 by finding values of \( \delta \) that correspond to \( \varepsilon = 0.5 \) and \( \varepsilon = 0.1 \).

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{t \to 2}\left( \frac{t^2 - 2}{t^3 - 3t + 5} \right)^2 \)

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion \( s = 2 \sin \pi t + 3 \cos \pi t \), where \( t \) is measured in seconds. (a) Find the average velocity during each time period: (i) \( [1, 2] \) (ii) \( [1, 1.1] \) (iii) \( [1, 1.01] \) (iv) \( [1, 1.001] \) (b) Estimate the instantaneous velocity of the particle when \( t =1 \).

(a) Find the slope of the tangent to the curve \( y = 3 + 4x^2 - 2x^3 \) at the point where \( x = a \). (b) Find equations of the tangent lines at the points \( (1, 5) \) and \( (2, 3) \). (c) Graph the curve and both tangents on a common screen.

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( f(0) = 3 \), \( \displaystyle \lim_{x \to 0^-} f(x) = 4 \), \( \displaystyle \lim_{x \to 0^+} f(x) = 2 \), \( \displaystyle \lim_{x \to -\infty} f(x) = -\infty \), \( \displaystyle \lim_{x \to 4^-} f(x) = -\infty \), \( \displaystyle \lim_{x \to 4^+} f(x) = \infty \), \( \displaystyle \lim_{x \to \infty} f(x) = 3 \)

The toll \( T \) charged for driving on a certain stretch of a toll road is 5 dollars except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is 7 dollars. (a) Sketch a graph of \( T \) as a function of the time \( t \), measured in hours past midnight. (b) Discuss the discontinuities of this function and their significance to someone who uses the road.

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{x \to 2}\sqrt{\frac{2x^2 + 1}{3x - 2}} \)

(a) Find the slope of the tangent to the curve \( y = 1/\sqrt{x} \) at the point where \( x = a \). (b) Find equations of the tangent lines at the points \( (1, 1) \) and \( (4, \frac{1}{2}) \). (c) Graph the curve and both tangents on a common screen.

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 3} f(x) = -\infty \), \( \displaystyle \lim_{x \to \infty} f(x) = 2 \), \( f(0) = 0 \), \( f \) is even

Explain why each function is continuous or discontinuous. (a) The temperature at a specific location as a function of time (b) The temperature at a specific time as a function of the distance due west from New York City (c) The altitude above sea level as a function of the distance due west from New York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time

(a) What is wrong with the following equation? $$ \frac {x^2 + x - 6}{x - 2} = x + 3 $$ (b) In view of part (a), explain why the equation $$ \lim_{x \to 2}\frac {x^2 + x - 6}{x - 2} = \lim_{x \to 2}(x + 3) $$ is correct.

Guess the value of the limit $$ \lim_{x \to \infty} \frac{x^2}{2^x} $$ by evaluating the function \( f(x) = x^2/2^x \) for \( x \) = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of \( f \) to support your guess.

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \( a \). \( f(x) = (x + 2x^3)^4, \hspace{5mm} a = -1 \)

A machinist is required to manufacture a circular metal disk with area 1000 \( cm^2 \). (a) What radius produces such a disk? (b) If the machinist is allowed an error tolerance of \( \pm 5 cm^2 \) in the area of the disk, how close to the ideal radius in part (a) must the machinist control the radius? (c) In terms of the \( \varepsilon \), \( \delta \) definition of \( \displaystyle \lim_{x \to a} f(x) = L \), what is \( x \)? What is \( f(x) \)? What is \( a \)? What is \( L \)? What value of \( \varepsilon \) is given? What is the corresponding value of \( \delta \)?

Evaluate the limit, if it exists. \( \displaystyle \lim_{x \to 5}\frac{x^2 - 6x + 5}{x - 5} \)

Sketch the graph of the function and use it to determine the values of \( a \) for which \( \displaystyle \lim_{x\to a}f(x) \) exists. $$ f(x) = \left\\{ \begin{array}{ll} 1 + x & \mbox {if \( x < -1 \)}\\\ x^2 & \mbox{if \( -1 \le x < 1\)}\\\ 2 - x & \mbox{if \( x \ge 1 \)} \end{array} \right.$$