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Chapter 2: Limits and The Derivative

Chapter 2 Review Problems

Review Problems

1.
What is the slope of the line through [latex](3,9)[/latex] and [latex](x, y)[/latex] for [latex]y = x^2[/latex] and [latex]x = 2.97?, \  x = 3.001?, \ x = 3+h[/latex]? What happens to this last slope when h is very small (close to [latex]0[/latex])?

2.

What is the slope of the line through [latex](–2,4)[/latex] and [latex](x, y)[/latex] for [latex]y = x^2[/latex] and [latex]x = –1.98?, \  x = –2.03?, \ x = –2+h[/latex]? What happens to this last slope when h is very small (close to [latex]0[/latex])?

3.

Fig. 36 shows the temperature during a day in Ames.
(a) What was the average change in temperature from [latex]9[/latex] am to [latex]1[/latex] pm?
(b) Estimate how fast the temperature was rising at [latex]10[/latex] am and at [latex]7[/latex] pm.

Graph showing temperature in degrees Fahrenheit (°F) over time. The x-axis marks time of day at 8 AM, noon, 3 PM, and 7 PM. The y-axis ranges from 60°F to 90°F. A red curve begins at about 60°F at 8 AM, peaks near 85°F between noon and 3 PM, and drops back to around 60°F by 7 PM.
Figure 36

4.

Fig. 37 shows the distance of a car from a measuring position located on the edge of a straight road.
(a) What was the average velocity of the car from [latex]t = 0[/latex] to [latex]t = 30[/latex] seconds?
(b) What was the average velocity of the car from [latex]t = 10[/latex] to t = 30 seconds?
(c) About how fast was the car traveling at [latex]t = 10[/latex] seconds? At [latex]t = 20[/latex] s? At [latex]t = 30[/latex] s?
(d) What does the horizontal part of the graph between [latex]t = 15[/latex] and [latex]t = 20[/latex] seconds mean?
(e) What does the negative velocity at [latex]t = 25[/latex] represent?

Graph showing distance in feet over time in seconds. The x-axis is labeled 'time (seconds)' and the y-axis is labeled 'distance (feet).' A blue curve with black data points starts at the origin, rises to about 100 feet at 10 seconds, continues to 200 feet at 15 seconds, levels off, then rises to 250 feet at 20 seconds. It dips slightly below 200 feet around 25 seconds before sharply increasing to nearly 300 feet at 30 seconds.
Figure 37

5.

Fig. 38 shows the distance of a car from a measuring position located on the edge of a straight road.
(a) What was the average velocity of the car from [latex]t = 0[/latex] to [latex]t = 20[/latex] seconds?
(b) What was the average velocity from [latex]t = 10[/latex] to [latex]t = 30[/latex] seconds?
(c) About how fast was the car traveling at [latex]t = 10[/latex] seconds? At [latex]t = 20[/latex] s? At [latex]t = 30[/latex] s?

A graph with the x-axis labeled "time (seconds)" and the y-axis labeled "distance (feet)". The graph shows a red curve that starts at the origin, rises to a peak of 300 feet at around 15 seconds, and then descends back down to 100 feet by 30 seconds. Black data points are plotted along the curve at intervals.
Figure 38

6.
Use the function in Fig. 39 to fill in the table and then graph [latex]m(x)[/latex].

 

A curve starts at the point (0, 1), rises to a peak around (1, 1.5), then goes down and crosses the x-axis at about (2, 0). It continues downward to a low point near (3, -1), and then rises again toward (4, -0.5).
Figure 39
x y= f(x) m(x) = the estimated slope of tangent line to y= f(x) at the point (x,y)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

 

7.

The graph of [latex]y = f(x)[/latex] is given in Fig. 40. Set up a table of values for [latex]x[/latex] and [latex]m(x)[/latex] (the slope of the line tangent to the graph of [latex]y=f(x)[/latex] at the point [latex](x,y)[/latex] ) and then graph the function [latex]m(x)[/latex].

The x-axis goes from 0 to 5, and the y-axis goes from 0 to 3. The curve begins around the point (0, 2.5), goes down to a low point near (2, 1), then rises to a high point around (4, 2), and finally goes down again toward (5, 1).
Figure 40

8.

(a) At what values of [latex]x[/latex] does the graph of [latex]f[/latex] in Fig. 41 have a horizontal tangent line?
(b) At what value(s) of [latex]x[/latex] is the value of [latex]f[/latex] the largest? Smallest?
(c) Sketch the graph of [latex]m(x)[/latex]= the slope of the line tangent to the graph of f at the point [latex](x,y)[/latex].

The x-axis goes from 0 to 5, and the y-axis goes from -1 to 2. The red curve starts just above the x-axis at x = 0, rises to a peak just below y = 2 around x = 1, dips below the x-axis to a low point near y = -1 around x = 3, then rises again to a peak slightly above y = 2 near x = 4.5, and begins to descend as it approaches x = 5
Figure 41

9.
(a) At what values of [latex]x[/latex] does the graph of [latex]g[/latex] in Fig. 42 have a horizontal tangent line?
(b) At what value(s) of [latex]x[/latex] is the value of [latex]g[/latex] the largest? Smallest?
(c) At what value(s) of [latex]x[/latex] is the slope of [latex]g[/latex] the largest? Smallest?

The graph's x-axis ranges from -1 to 5, and the y-axis ranges from -1 to 2. The blue curve starts at the point (0, 2), dips down to about (1, 1), rises to a peak near (2, 2), dips again slightly above (3, 1), and then rises steadily toward (5, 2).
Figure 42

10.
Match the situation descriptions with the corresponding time–velocity graphs in Fig. 43.
(a) A car quickly leaving from a stop sign.
(b) A car sedately leaving from a stop sign.
(c) A student bouncing on a trampoline.
(d) A ball thrown straight up.
(e) A student confidently striding across campus to take a calculus test.
(f) An unprepared student walking across campus to take a calculus test.

The image shows six different graphs labeled (a) through (f), each representing a different type of function or trend: Graph (a): A blue curve that starts low, increases steadily, and then levels off. Graph (b): A red straight line with a negative slope. Graph (c): A blue horizontal line. Graph (d): A red curve with multiple oscillations. Graph (e): A blue sinusoidal wave with several peaks and troughs. Graph (f): A red curve that starts low and increases steadily without leveling off.
Figure 43

11.

Fig. 44 shows the temperature during a summer day in Chicago. Sketch the graph of the rate at which the temperature is changing. (This is just the graph of the slopes of the lines that are tangent to the temperature graph.)

The graph shows temperature changes over the course of a day. The x-axis represents time, with labels at 6 AM, noon, and 6 PM. The y-axis shows temperature in degrees Fahrenheit, ranging from 70°F to 100°F. The temperature rises from 6 AM to a peak slightly above 90°F around noon, then dips slightly, rises again, and gradually decreases toward 6 PM.
Figure 44

12.
Fig. 45 shows six graphs, three of which are derivatives of the other three. Match the functions with their derivatives.

The image contains six graphs labeled (a) through (f), each showing a different mathematical function or relationship: Graph (a): A smooth curve with one peak and one trough. Graph (b): Two linear segments—one ends with an open circle, and the other begins with a closed circle. Graph (c): An increasing curve that gradually levels off. Graph (d): A decreasing curve that gradually levels off. Graph (e): A sinusoidal-like wave with repeating peaks and troughs. Graph (f): Two horizontal lines—one has an open circle in the middle, and the other intersects it perpendicularly.
Figure 45

13.
Match the graphs of the three functions in Fig. 46 with the graphs of their derivatives.

The image shows six graphs arranged in two rows and three columns. The top row displays three functions labeled f, g, and h. The bottom row shows the corresponding derivatives of these functions. First column: Function f is a piecewise linear graph that increases from the point (1, -1) to (2, 1), then decreases to (3, -1). Its derivative, shown below, is a constant negative slope. Second column: Function g is a smooth curve that rises to a peak at x = 2 and then symmetrically decreases. Its derivative starts at zero, increases to a peak at x = 2, and then returns to zero. Third column: Function h is an increasing curve starting from x = 0. Its derivative is shown as two horizontal lines with constant positive slopes on either side of an open circle at x = 0.
Figure 46

14. Use the information in Fig. 47 to plot the values of the functions [latex]f + g, \ f.g[/latex] and [latex]f/g[/latex] and their derivatives at [latex]x = 1[/latex], [latex]2[/latex], and [latex]3[/latex].

The image shows two side-by-side graphs. The left graph is labeled "y = f(x)" and is drawn in blue. It starts at y = 3 when x = 0, decreases to a minimum near y = 1 just before x = 2, and then increases again as x approaches 3. The right graph is labeled "y = g(x)" and is drawn in red. It starts at y = 1 when x = 0, rises to a peak near y = 3 just after x = 1, dips to a local minimum around y = 2 at x ≈ 2, then rises again before decreasing as x nears 3. Both graphs include grid lines with tick marks at intervals of 1 along both the x- and y-axes, ranging from 0 to 3.
Figure 47

15.

Calculate [latex]\frac{d}{dx} \bigg( (x-5)(3x+1) \bigg)[/latex] by (a) using the product rule and (b) expanding the product and then differentiating. Verify that both methods give the same result.

16.

If the product of [latex]f[/latex] and [latex]g[/latex] is a constant ([latex]f(x) ∙ g(x) = k[/latex] for all [latex]x[/latex]), then how are [latex]\frac{\frac{d}{dx}\bigg( f(x) \bigg)}{f(x)}[/latex] and [latex]\frac{\frac{d}{dx}\bigg( g(x) \bigg)}{g(x)}[/latex] related?

17.

If the quotient of [latex]f[/latex] and [latex]g[/latex] is a constant (for all [latex]x[/latex]), then how are [latex]g . f '[/latex] and [latex]f . g '[/latex] related?

 

In problems 20–25, (a) calculate [latex]f '(1)[/latex] and (b) determine when [latex]f '(x) = 0[/latex].

18.

[latex]f(x) = x^2 – 5x + 13[/latex]

19.

[latex]f(x) = 5x^2 – 40x + 73[/latex]

20.

[latex]f(x) = x^3 + 9x^2 + 6[/latex]

21.

[latex]f(x) = x^3 + 3x^2 + 3x – 1[/latex]

22.

[latex]f(x) = x^3 + 2x^2 + 2x – 1[/latex]

23.

[latex]f(x) = \frac{7x}{x^2+4}[/latex]

24.

Determine [latex]\frac{d}{dx}\bigg( (x^2+1)(7x-3) \bigg)[/latex] and [latex]\frac{d}{dt} \bigg( \frac{3t-2}{5t+1} \bigg)[/latex].

25.

Find (a) [latex]\frac{d}{dx} \bigg( x^3e^x \bigg)[/latex] and (b) [latex]\frac{d}{dx} \bigg( e^x \bigg)^3[/latex].

 

26.

Find (a) [latex]\frac{d}{dt} \bigg( te^t \bigg)[/latex] and (b) [latex]\frac{d}{dx} \bigg( e^x \bigg)^3[/latex].

 

27:

Where do [latex]f(x) = x^2 – 10x + 3[/latex] and [latex]g(x) = x^3 – 12x[/latex] have horizontal tangent lines?

Brief Answers to Selected Exercises

1.

[latex]m = \frac{y-9}{x-3}[/latex]. If [latex]x = 2.97[/latex], then [latex]m = \frac{-0.1791-9}{-0.03} = 5.97[/latex]. If [latex]x = 3.001[/latex], then [latex]m = \frac{0.00601}{0.001} = 6.001[/latex].
If [latex]x = 3+h[/latex], then [latex]m = \frac{(3+h)^2-9}{((3+h)-3} = \frac{9+6h+h^2-9}{h} = 6+h[/latex]. When [latex]h[/latex] is very small (close to [latex]0[/latex]), [latex]6 + h[/latex] is very close to [latex]6[/latex].

3.

All of these answers are approximate. Your answers should be close to these numbers.

  1. [latex]\text{change} \approx \frac{80^o-64^o}{1 \text{pm} - 9 \text{am}} = \frac{16^o}{4 \text{hours}} = 4^o \text{per hour}[/latex].
  2. Use the slope of the tangent line. At [latex]10\ \text{am}[/latex], temperature was rising about [latex]5^{o}\ \text{per hour}[/latex]. At [latex]7 \text{pm}[/latex], temperature was rising about [latex]-10^{o}\ \text{per hour}[/latex] (falling about [latex]-10^{o}\ \text{per hour}[/latex]).

5.

 

All of these answers are approximate. Your answers should be close to these numbers.

  1. [latex]\text{average velocity} \approx \frac{300 \ \text{ft} - 0 \ \text{ft}}{20 \ \text{sec} - 0 \ \text{sec}}= -1\ \text{ feet per hour}[/latex].
  2. [latex]\text{average velocity} \approx \frac{100 \ \text{ft} - 200 \ \text{ft}}{30 \ \text{sec} - 20 \ \text{sec}}= - 5\ \text{ feet per hour}[/latex].
  3. At [latex]t = 10[/latex] seconds, [latex]\text{velocity} \approx 30 \ \text{ feet per second}[/latex] (between [latex]20[/latex] and [latex]35 \ \text{ft/s}[/latex]). At [latex]t = -1[/latex] seconds, [latex]\text{velocity} \approx 30\ \text{ feet per second}[/latex]. At [latex]t = 30[/latex] seconds,  [latex]\text{velocity} \approx -40\ \text{ feet per second}[/latex].

7.

Below is a graph of [latex]m(x)[/latex]. Approximate values of [latex]m(x)[/latex] are in the table.

The image contains two graphs. The top graph, labeled "y = f(x)", is a blue curve that starts at y = 2 when x = 0, dips to a minimum near the point (2, 0.5), rises to a peak around (4, 1.5), and then decreases toward y = 1 as x approaches 5. The bottom graph is labeled "approximate graph of m(x) = slopes of f(x)" and is a red curve. It starts at y = -1 when x = 0, rises to a maximum near (3, 1), and then decreases back toward y = -1 as x approaches 5.
x y= f(x) m(x) = the estimated slope of tangent line to y= f(x) at the point (x,y)
0 2 -1
1 1 -1
2 1/3 0
3 1 1
4 3/2 1/2
5 1 -2

9.

a. The graph of [latex]g[/latex] has a horizontal tangent line at about [latex]x = 1, x = 2[/latex], and [latex]x = 3[/latex].

b. [latex]g[/latex] is largest when [latex]x = 2[/latex] and for [latex]x = -0.5[/latex] and [latex]x = 6[/latex] (the edges of the graph). [latex]g[/latex] is smallest when [latex]x = 1[/latex] and [latex]x = 3[/latex].

c. The slope of [latex]g[/latex] is the largest when [latex]x = \   \text{about} \  1.5[/latex], when the graph of [latex]g[/latex] is steepest while increasing. The slope of [latex]g[/latex] is smallest when [latex]x = \ \text{about} \  2.5[/latex], when the graph of [latex]g[/latex] is steepest while decreasing.

 

11.
Fig. 2 is a graph of the approximate rate of temperature change (slope).

The image contains two graphs. The top graph shows the temperature in degrees Fahrenheit over time, labeled "y = f(x)". The y-axis is labeled "temperature (°F)", and the x-axis is marked with time points at 6 AM, noon, and 6 PM. The blue curve rises from 6 AM to noon, then fluctuates until 6 PM. The bottom graph shows the rate of temperature change over time, labeled "approximate slopes of f". The y-axis is labeled "rate of temp change (degrees/hour)", and the x-axis also shows 6 AM, noon, and 6 PM. The red curve starts at zero at 6 AM, dips below zero before noon, rises above zero after noon, and then fluctuates around zero until 6 PM.

13.

The left derivative graph is [latex]g’[/latex]. The middle derivative graph is [latex]h'[/latex]. The right derivative graph is [latex]f’[/latex].

 

15.

a. [latex]\frac{d}{dx} \bigg( (x-5)(3x+7) \bigg) = (1)(3x+7)+(x-5)(3)=6x-8[/latex]

b.  [latex]\frac{d}{dx} \bigg( (x-5)(3x+7) \bigg) =\frac{d}{dx} (3x^2-8x-35)=6x-8[/latex]

 

17.

If [latex]\frac{f(x)}{g(x)}= k[/latex], then [latex]\frac{d}{dx}\bigg( \frac{f(x)}{g(x)} \bigg) =0[/latex]. So the numerator of the rational function we’d get from the quotient rule must be zero; that is,  [latex]f'g-fg' = 0[/latex]. So [latex]f'g[/latex] and [latex]fg'[/latex] are equal. (Another way to think about this: If [latex]\frac{f(x)}{g(x)}= k[/latex], then [latex]f(x)=kg(x)[/latex] and [latex]f'(x) = kg'g = kgg' = fg'[/latex].)

 

19.

If [latex]f(x) = x^2-5x+13[/latex], so [latex]f'(x) = 2x-5, \ f'(1) = 2-5=-3[/latex], and [latex]f'(x) = 2x-5 =0[/latex] when [latex]x=2.5[/latex].

 

21.

[latex]f(x) = x^3+3x^2+3x-1[/latex], so [latex]f'(x) = 3x^2+6x+3, f'(1) = 12[/latex], and [latex]f'(x) = 0[/latex] when [latex]x=-1[/latex].

23.

[latex]f(x) = \frac{7x}{x^2+4}[/latex], so [latex]f'(x) = \frac{(70)(x^2-4)-(7x)(2x)}{(x^2+4)^2} = \frac{-7x+28}{(x^2+4)^2}, f'(1) = \frac{21}{25}[/latex], and [latex]f(x) =0[/latex] when [latex]x=4[/latex].

25.

a. [latex]\frac{d}{dx}(x^3e^x)= (3x^2)(e^x) + (x^3)(e^x)[/latex].
b. [latex]\frac{d}{dx}(e^x)^3) = 3(e^x)^2(e^x) = 3e^{3x}[/latex].
Alternatively, [latex]\frac{d}{dx}(e^x)^3) = \frac{d}{dx}(e^{3x}) = 3e^{3x}[/latex].

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