Stewart 9e Section 2.1: Derivatives and Rates of Change

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Stewart 9e Section 2.1: Derivatives and Rates of Change 0/68
1 Exercise - Secant and tangent slopes · Level 1
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 \(P\).

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2 Exercise - Zooming in on a curve · Level 1
Graph the curve \(y = \sin x\) in the viewing rectangles \([-2, 2]\) by \([-2, 2]\), \([-1, 1]\) by \([-1, 1]\), and \([-0.5, 0.5]\) by \([-0.5, 0.5]\). What do you notice about the curve as you zoom in toward the origin?
3 Exercise - Tangent slope two ways · Level 2
(a) Find the slope of the tangent line to the parabola \(y = x^2 + 3 x\) at the point \((-1, -2)\) (i) using Definition 1, (ii) using Equation 2.
(b) Find an equation of the tangent line in part (a).
(c) Graph the parabola and the tangent line; zoom in toward the point \((-1, -2)\) to verify.

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4 Exercise - Tangent slope two ways · Level 2
(a) Find the slope of the tangent line to the curve \(y = x^3 + 1\) at the point \((1, 2)\) (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 \((1, 2)\).

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5 Exercise - Tangent line · Level 2
Find an equation of the tangent line to the curve \(y = 2 x^2 - 5 x + 1\) at the point \((3, 4)\).
6 Exercise - Tangent line · Level 2
Find an equation of the tangent line to the curve \(y = x^2 - 2 x^3\) at the point \((1, -1)\).
7 Exercise - Tangent line · Level 2
Find an equation of the tangent line to the curve \(y = \dfrac{x + 2}{x - 3}\) at the point \((2, -4)\).
8 Exercise - Tangent line · Level 2
Find an equation of the tangent line to the curve \(y = \sqrt{1 - 3 x}\) at the point \((-1, 2)\).
9 Exercise - General tangent slope · Level 3
(a) Find the slope of the tangent to the curve \(y = 3 + 4 x^2 - 2 x^3\) 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.

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10 Exercise - General tangent slope · Level 2
(a) Find the slope of the tangent to the curve \(y = 2 \sqrt{x}\) at the point where \(x = a\).
(b) Find equations of the tangent lines at the points \((1, 2)\) and \((9, 6)\).
(c) Graph the curve and both tangents on a common screen.

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11 Exercise - Velocity from position · Level 2
A cliff diver plunges from a height of \(100\) ft above the water surface. The distance the diver falls in \(t\) seconds is given by the function \(d(t) = 16 t^2\) ft.
(a) After how many seconds will the diver hit the water?
(b) With what velocity does the diver hit the water?

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12 Exercise - Velocity from position · Level 3
If a rock is thrown upward on the planet Mars with a velocity of \(10\) m/s, its height (in meters) after \(t\) seconds is given by \(H = 10 t - 1.86 t^2\).
(a) Find the velocity of the rock after one second.
(b) Find the velocity of the rock when \(t = a\).
(c) When will the rock hit the surface?
(d) With what velocity will the rock hit the surface?

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13 Exercise - Velocity from position · Level 2
The displacement (in meters) of a particle moving in a straight line is given by the equation of motion \(s = 1/t^2\), where \(t\) is measured in seconds. Find the velocity of the particle at times \(t = a\), \(t = 1\), \(t = 2\), and \(t = 3\).
14 Exercise - Average vs instantaneous velocity · Level 3
The displacement (in feet) of a particle moving in a straight line is given by \(s = \dfrac{1}{2} t^2 - 6 t + 23\), where \(t\) is measured in seconds.
(a) Find the average velocity over each time interval: (i) \([4, 8]\) (ii) \([6, 8]\) (iii) \([8, 10]\) (iv) \([8, 12]\).
(b) Find the instantaneous velocity when \(t = 8\).
(c) Draw the graph of \(s\) as a function of \(t\) and draw the secant lines whose slopes are the average velocities in part (a). Then draw the tangent line whose slope is the instantaneous velocity in part (b).

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15 Exercise - Reading a position graph · Level 2
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(a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still?
(b) Draw a graph of the velocity function.

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16 Exercise - Comparing runners · Level 2
Shown are graphs of the position functions of two runners, A and B, who run a 100-meter race and finish in a tie.
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(a) Describe and compare how the runners run the race.
(b) At what time is the distance between the runners the greatest?
(c) At what time do they have the same velocity?

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17 Exercise - Ordering derivative values · Level 2
For the function \(g\) whose graph is given, arrange the following numbers in increasing order and explain your reasoning: \(0, \quad g'(-2), \quad g'(0), \quad g'(2), \quad g'(4)\).
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18 Exercise - Reading a function graph · Level 3
The graph of a function \(f\) is shown.
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(a) Find the average rate of change of \(f\) on the interval \([20, 60]\).
(b) Identify an interval on which the average rate of change of \(f\) is \(0\).
(c) Compute \(\dfrac{f(40) - f(10)}{40 - 10}\). What does this value represent geometrically?
(d) Estimate the value of \(f'(50)\).
(e) Is \(f'(10) > f'(30)\)?
(f) Is \(f'(60) > \dfrac{f(80) - f(40)}{80 - 40}\)? Explain.

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19 Exercise - Derivative from Definition 4 · Level 2
Use Definition 4 to find \(f'(a)\) at the given number \(a\): \(f(x) = \sqrt{4 x + 1}\), \(a = 6\).
20 Exercise - Derivative from Definition 4 · Level 2
Use Definition 4 to find \(f'(a)\) at the given number \(a\): \(f(x) = 5 x^4\), \(a = -1\).
21 Exercise - Derivative from Equation 5 · Level 3
Use Equation 5 to find \(f'(a)\) at the given number \(a\): \(f(x) = \dfrac{x^2}{x + 6}\), \(a = 3\).
22 Exercise - Derivative from Equation 5 · Level 3
Use Equation 5 to find \(f'(a)\) at the given number \(a\): \(f(x) = \dfrac{1}{\sqrt{2 x + 2}}\), \(a = 1\).
23 Exercise - Derivative at a general $a$ · Level 2
Find \(f'(a)\) for \(f(x) = 2 x^2 - 5 x + 3\).
24 Exercise - Derivative at a general $a$ · Level 2
Find \(f'(a)\) for \(f(t) = t^3 - 3 t\).
25 Exercise - Derivative at a general $a$ · Level 2
Find \(f'(a)\) for \(f(t) = \dfrac{1}{t^2 + 1}\).
26 Exercise - Derivative at a general $a$ · Level 3
Find \(f'(a)\) for \(f(x) = \dfrac{x}{1 - 4 x}\).
27 Exercise - Tangent from given values · Level 1
Find an equation of the tangent line to the graph of \(y = B(x)\) at \(x = 6\) if \(B(6) = 0\) and \(B'(6) = -\dfrac{1}{2}\).
28 Exercise - Tangent from given values · Level 1
Find an equation of the tangent line to the graph of \(y = g(x)\) at \(x = 5\) if \(g(5) = -3\) and \(g'(5) = 4\).
29 Exercise - Tangent via derivative · Level 2
If \(f(x) = 3 x^2 - x^3\), find \(f'(1)\) and use it to find an equation of the tangent line to the curve \(y = 3 x^2 - x^3\) at the point \((1, 2)\).
30 Exercise - Tangent via derivative · Level 2
If \(g(x) = x^4 - 2\), find \(g'(1)\) and use it to find an equation of the tangent line to the curve \(y = x^4 - 2\) at the point \((1, -1)\).
31 Exercise - Tangent via derivative · Level 3
(a) If \(F(x) = \dfrac{5 x}{1 + x^2}\), find \(F'(2)\) and use it to find an equation of the tangent line to the curve \(y = \dfrac{5 x}{1 + x^2}\) at the point \((2, 2)\).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

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32 Exercise - Tangent via derivative · Level 3
(a) If \(G(x) = 4 x^2 - x^3\), find \(G'(a)\) and use it to find equations of the tangent lines to the curve \(y = 4 x^2 - x^3\) at the points \((2, 8)\) and \((3, 9)\).
(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

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33 Exercise - Reading $f$ and $f'$ from a tangent · Level 1
If an equation of the tangent line to the curve \(y = f(x)\) at the point where \(a = 2\) is \(y = 4 x - 5\), find \(f(2)\) and \(f'(2)\).
34 Exercise - Reading $f$ and $f'$ from a tangent · Level 1
If the tangent line to \(y = f(x)\) at \((4, 3)\) passes through the point \((0, 2)\), find \(f(4)\) and \(f'(4)\).
35 Exercise - Velocity and speed · Level 2
A particle moves along a straight line with equation of motion \(s = f(t)\), where \(s\) is measured in meters and \(t\) in seconds. Find the velocity and the speed when \(t = 4\) for \(f(t) = 80 t - 6 t^2\).
36 Exercise - Velocity and speed · Level 2
A particle moves along a straight line with equation of motion \(s = f(t)\). Find the velocity and the speed when \(t = 4\) for \(f(t) = 10 + \dfrac{45}{t + 1}\).
37 Exercise - Qualitative graph (cooling) · Level 2
A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?
38 Exercise - Estimating a rate from a graph · Level 2
A roast turkey is taken from an oven when its temperature has reached \(185^{\circ} F\) and is placed on a table in a room where the temperature is \(75^{\circ} F\). The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.
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39 Exercise - Sketch given derivative info · Level 2
Sketch the graph of a function \(f\) for which \(f(0) = 0\), \(f'(0) = 3\), \(f'(1) = 0\), and \(f'(2) = -1\).
40 Exercise - Sketch given derivative info · Level 3
Sketch the graph of a function \(g\) for which \(g(0) = g(2) = g(4) = 0\), \(g'(1) = g'(3) = 0\), \(g'(0) = g'(4) = 1\), \(g'(2) = -1\), \(\operatorname*{lim}\limits_{x\rightarrow 5^-} g(x) = \infty\), and \(\operatorname*{lim}\limits_{x\rightarrow -1^+} g(x) = -\infty\).
41 Exercise - Sketch given derivative info · Level 2
Sketch the graph of a function \(g\) that is continuous on its domain \((-5, 5)\) and where \(g(0) = 1\), \(g'(0) = 1\), \(g'(-2) = 0\), \(\operatorname*{lim}\limits_{x\rightarrow -5^+} g(x) = \infty\), and \(\operatorname*{lim}\limits_{x\rightarrow 5^-} g(x) = 3\).
42 Exercise - Sketch given derivative info · Level 3
Sketch the graph of a function \(f\) where the domain is \((-2, 2)\), \(f'(0) = -2\), \(\operatorname*{lim}\limits_{x\rightarrow 2^-} f(x) = \infty\), \(f\) is continuous at all numbers in its domain except \(\pm 1\), and \(f\) is odd.
43 Exercise - Identify $f$ and $a$ from a limit · Level 2
Each limit represents the derivative of some function \(f\) at some number \(a\). State such an \(f\) and \(a\): \(\operatorname*{lim}\limits_{h\rightarrow 0} \dfrac{\sqrt{9 + h} - 3}{h}\).
44 Exercise - Identify $f$ and $a$ from a limit · Level 2
Identify \(f\) and \(a\): \(\operatorname*{lim}\limits_{h\rightarrow 0} \dfrac{2^{3 + h} - 8}{h}\).
45 Exercise - Identify $f$ and $a$ from a limit · Level 2
Identify \(f\) and \(a\): \(\operatorname*{lim}\limits_{x\rightarrow 2} \dfrac{x^6 - 64}{x - 2}\).
46 Exercise - Identify $f$ and $a$ from a limit · Level 2
Identify \(f\) and \(a\): \(\operatorname*{lim}\limits_{x\rightarrow \dfrac{1}{4}} \dfrac{\dfrac{1}{x} - 4}{x - \dfrac{1}{4}}\).
47 Exercise - Identify $f$ and $a$ from a limit · Level 2
Identify \(f\) and \(a\): \(\operatorname*{lim}\limits_{h\rightarrow 0} \dfrac{\tan\left(\dfrac{\pi}{4} + h\right) - 1}{h}\).
48 Exercise - Identify $f$ and $a$ from a limit · Level 2
Identify \(f\) and \(a\): \(\operatorname*{lim}\limits_{\theta\rightarrow \dfrac{\pi}{6}} \dfrac{\sin \theta - \dfrac{1}{2}}{\theta - \dfrac{\pi}{6}}\).
49 Exercise - Marginal cost · Level 3
The cost (in dollars) of producing \(x\) units of a certain commodity is \(C(x) = 5000 + 10 x + 0.05 x^2\).
(a) Find the average rate of change of \(C\) with respect to \(x\) when the production level is changed (i) from \(x = 100\) to \(x = 105\) and (ii) from \(x = 100\) to \(x = 101\).
(b) Find the instantaneous rate of change of \(C\) with respect to \(x\) when \(x = 100\) (the marginal cost).

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50 Exercise - Interpreting a derivative · Level 2
Let \(H(t)\) be the daily cost (in dollars) to heat an office building when the outside temperature is \(t\) degrees Fahrenheit.
(a) What is the meaning of \(H'(58)\)? What are its units?
(b) Would you expect \(H'(58)\) to be positive or negative? Explain.

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51 Exercise - Interpreting a derivative (mining) · Level 2
The cost of producing \(x\) ounces of gold from a new gold mine is \(C = f(x)\) dollars.
(a) What is the meaning of the derivative \(f'(x)\)? What are its units?
(b) What does the statement \(f'(800) = 17\) mean?
(c) Do you think the values of \(f'(x)\) will increase or decrease in the short term? What about the long term? Explain.

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52 Exercise - Interpreting a derivative (demand) · Level 2
The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of \(p\) dollars per pound is \(Q = f(p)\).
(a) What is the meaning of the derivative \(f'(8)\)? What are its units?
(b) Is \(f'(8)\) positive or negative? Explain.

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53 Exercise - Interpreting a derivative (oxygen solubility) · Level 2
The quantity of oxygen that can dissolve in water depends on the temperature of the water. The graph shows how oxygen solubility \(S\) varies as a function of the water temperature \(T\).
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(a) What is the meaning of the derivative \(S'(T)\)? What are its units?
(b) Estimate the value of \(S'(16)\) and interpret it.

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54 Exercise - Interpreting a derivative (salmon) · Level 2
The graph shows the influence of the temperature \(T\) on the maximum sustainable swimming speed \(S\) of Coho salmon.
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(a) What is the meaning of the derivative \(S'(T)\)? What are its units?
(b) Estimate the values of \(S'(15)\) and \(S'(25)\) and interpret them.

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55 Exercise - Rate of change from a table · Level 3
Researchers measured the average blood alcohol concentration \(C(t)\) of eight men starting one hour after consumption of 30 mL of ethanol. \(t = 1.0\) h, \(C(t) = 0.033\) g/dL. \(t = 1.5\) h, \(C(t) = 0.024\) g/dL. \(t = 2.0\) h, \(C(t) = 0.018\) g/dL. \(t = 2.5\) h, \(C(t) = 0.012\) g/dL. \(t = 3.0\) h, \(C(t) = 0.007\) g/dL.
(a) Find the average rate of change of \(C\) with respect to \(t\) over each time interval: (i) \([1.0, 2.0]\) (ii) \([1.5, 2.0]\) (iii) \([2.0, 2.5]\) (iv) \([2.0, 3.0]\). Include the units.
(b) Estimate the instantaneous rate of change at \(t = 2\) and interpret your result. What are the units?

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56 Exercise - Rate of change from a table · Level 3
The number \(N\) of locations of a popular coffeehouse chain is given in the table (as of October 1 each year): Year \(2008\): \(N = 16{,}680\) Year \(2010\): \(N = 16{,}858\) Year \(2012\): \(N = 18{,}066\) Year \(2014\): \(N = 21{,}366\) Year \(2016\): \(N = 25{,}085\)
(a) Find the average rate of growth (i) from 2008 to 2010 and (ii) from 2010 to 2012. Include the units. What can you conclude?
(b) Estimate the instantaneous rate of growth in 2010 by taking the average of two average rates of change. What are its units?
(c) Estimate the instantaneous rate of growth in 2010 by measuring the slope of a tangent.

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57 Exercise - Existence of $f'(0)$ · Level 3
Determine whether \(f'(0)\) exists for \(f(x) = \begin{cases} x \sin\left(\dfrac{1}{x}\right) & \quad \text{if } x \neq 0 \\ 0 & \quad \text{if } x = 0 \end{cases}\).
58 Exercise - Existence of $f'(0)$ · Level 3
Determine whether \(f'(0)\) exists for \(f(x) = \begin{cases} x^2 \sin\left(\dfrac{1}{x}\right) & \quad \text{if } x \neq 0 \\ 0 & \quad \text{if } x = 0 \end{cases}\).
59 Exercise - Visual derivative estimate · Level 3
(a) Graph the function \(f(x) = \sin x - \dfrac{1}{1000} \sin(1000 x)\) in the viewing rectangle \([-2 \pi, 2 \pi]\) by \([-4, 4]\). What slope does the graph appear to have at the origin?
(b) Zoom in to the viewing window \([-0.4, 0.4]\) by \([-0.25, 0.25]\) and estimate the value of \(f'(0)\). Does this agree with your answer from part (a)?
(c) Now zoom in to the viewing window \([-0.008, 0.008]\) by \([-0.005, 0.005]\). Do you wish to revise your estimate for \(f'(0)\)?

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60 Exercise - Symmetric difference quotient · Level 3
In Example 8 we approximated an instantaneous rate of change by averaging two average rates of change. An alternative is to use a single average rate of change over an interval centered at the desired value. We define the symmetric difference quotient of \(f\) at \(x = a\) on \([a - d, a + d]\) as \(\dfrac{f(a + d) - f(a - d)}{2 d}\).
(a) Compute the symmetric difference quotient for the function \(D\) in Example 8 on the interval \([2004, 2012]\) and verify that your result agrees with the estimate for \(D'(2008)\) computed in that example.
(b) Show that the symmetric difference quotient of a function \(f\) at \(x = a\) is equivalent to averaging the average rates of change of \(f\) over the intervals \([a - d, a]\) and \([a, a + d]\).
(c) Use a symmetric difference quotient to estimate \(f'(1)\) for \(f(x) = x^3 - 2 x^2 + 2\) with \(d = 0.4\). Draw a graph of \(f\) along with secant lines corresponding to average rates of change over the intervals \([1 - d, 1]\), \([1, 1 + d]\), and \([1 - d, 1 + d]\). Which of these secant lines appears to have slope closest to that of the tangent line at \(x = 1\)?

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61 Example - Tangent line to parabola · Level 2
Find an equation of the tangent line to the parabola \(y = x^2\) at the point \(P(1, 1)\).
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62 Example - Tangent line to hyperbola · Level 2
Find an equation of the tangent line to the hyperbola \(y = \dfrac{3}{x}\) at the point \((3, 1)\).
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63 Example - Instantaneous velocity · Level 3
Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. The distance (in meters) fallen after \(t\) seconds is \(s = f(t) = 4.9 t^2\).
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(a) What is the velocity of the ball after 5 seconds?
(b) How fast is the ball traveling when it hits the ground?

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64 Example - Derivative from Definition 4 · Level 2
Use Definition 4 to find the derivative of the function \(f(x) = x^2 - 8x + 9\) at the numbers (a) \(2\) and (b) \(a\).
65 Example - Derivative from Equation 5 · Level 3
Use Equation 5 to find the derivative of the function \(f(x) = \dfrac{1}{\sqrt{x}}\) at the number \(a\) (where \(a > 0\)).
66 Example - Tangent line via derivative · Level 2
Find an equation of the tangent line to the parabola \(y = x^2 - 8x + 9\) at the point \((3, -6)\).
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67 Example - Interpreting a derivative (marginal cost) · Level 2
A manufacturer produces bolts of a fabric with a fixed width. The cost of producing \(x\) yards of this fabric is \(C = f(x)\) dollars.
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(a) What is the meaning of the derivative \(f'(x)\)? What are its units?
(b) In practical terms, what does it mean to say that \(f'(1000) = 9\)?
(c) Which do you think is greater, \(f'(50)\) or \(f'(500)\)? What about \(f'(5000)\)?

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68 Example - Estimating a derivative from a table · Level 3
Let \(D(t)\) be the US national debt at time \(t\). The table gives approximate values of this function by providing end-of-year estimates, in billions of dollars, from 2000 to 2016. \(t = 2000\): \(D(t) = 5662.2\) \(t = 2004\): \(D(t) = 7596.1\) \(t = 2008\): \(D(t) = 10{,}699.8\) \(t = 2012\): \(D(t) = 16{,}432.7\) \(t = 2016\): \(D(t) = 19{,}976.8\) Interpret and estimate the value of \(D'(2008)\).

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