Stewart Precalc 6e Section 13.3: Tangent Lines and Derivatives

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Stewart Precalc 6e Section 13.3: Tangent Lines and Derivatives 0/41
1 Concept · Level 1
The derivative of a function \(f\) at a number \(a\) is \(f'(a) = \) _____ if the limit exists. The derivative \(f'(a)\) is the _____ of the tangent line to the curve \(y = f(x)\) at the point ( _____, _____ ).
2 Concept · Level 1
If \(y = f(x)\), the average rate of change of \(f\) between the numbers \(x\) and \(a\) is _____. The limit of the average rates of change as \(x\) approaches \(a\) is the _____ rate of change of \(y\) with respect to \(x\) at \(x = a\); this is also the derivative \(f'(\) _____ \()\).
3 Slope of Tangent Line · Level 1
Find the slope of the tangent line to the graph of \(f\) at the given point: \(f(x) = 3 x + 4\) at \((1, 7)\).
4 Slope of Tangent Line · Level 1
Find the slope of the tangent line to the graph of \(f\) at the given point: \(f(x) = 5 - 2 x\) at \((-3, 11)\).
5 Slope of Tangent Line · Level 2
Find the slope of the tangent line to the graph of \(f\) at the given point: \(f(x) = 4 x^2 - 3 x\) at \((-1, 7)\).
6 Slope of Tangent Line · Level 2
Find the slope of the tangent line to the graph of \(f\) at the given point: \(f(x) = 1 + 2 x - 3 x^2\) at \((1, 0)\).
7 Slope of Tangent Line · Level 2
Find the slope of the tangent line to the graph of \(f\) at the given point: \(f(x) = 2 x^3\) at \((2, 16)\).
8 Slope of Tangent Line · Level 2
Find the slope of the tangent line to the graph of \(f\) at the given point: \(f(x) = \dfrac{6}{x + 1}\) at \((2, 2)\).
9 Equation of Tangent Line · Level 2
Find an equation of the tangent line to the curve at the given point: \(y = x + x^2\) at \((-1, 0)\).
10 Equation of Tangent Line · Level 3
Find an equation of the tangent line to the curve at the given point: \(y = 2 x - x^3\) at \((1, 1)\).
11 Equation of Tangent Line · Level 3
Find an equation of the tangent line to the curve at the given point: \(y = \dfrac{x}{x - 1}\) at \((2, 2)\).
12 Equation of Tangent Line · Level 3
Find an equation of the tangent line to the curve at the given point: \(y = \dfrac{1}{x^2}\) at \((-1, 1)\).
13 Equation of Tangent Line · Level 3
Find an equation of the tangent line to the curve at the given point: \(y = \sqrt{x + 3}\) at \((1, 2)\).
14 Equation of Tangent Line · Level 3
Find an equation of the tangent line to the curve at the given point: \(y = \sqrt{1 + 2 x}\) at \((4, 3)\).
15 Derivative at a Number · Level 2
Find the derivative of the function at the given number: \(f(x) = 1 - 3 x^2\) at 2.
16 Derivative at a Number · Level 2
Find the derivative of the function at the given number: \(f(x) = 2 - 3 x + x^2\) at \(-1\).
17 Derivative at a Number · Level 2
Find the derivative of the function at the given number: \(g(x) = x^4\) at 1.
18 Derivative at a Number · Level 2
Find the derivative of the function at the given number: \(g(x) = 2 x^2 + x^3\) at 1.
19 Derivative at a Number · Level 3
Find the derivative of the function at the given number: \(F(x) = \dfrac{1}{\sqrt{x}}\) at 4.
20 Derivative at a Number · Level 2
Find the derivative of the function at the given number: \(G(x) = 1 + 2 \sqrt{x}\) at 4.
21 Derivative f'(a) · Level 3
Find \(f'(a)\), where \(a\) is in the domain of \(f\): \(f(x) = x^2 + 2 x\).
22 Derivative f'(a) · Level 3
Find \(f'(a)\), where \(a\) is in the domain of \(f\): \(f(x) = -\dfrac{1}{x^2}\).
23 Derivative f'(a) · Level 3
Find \(f'(a)\), where \(a\) is in the domain of \(f\): \(f(x) = \dfrac{x}{x + 1}\).
24 Derivative f'(a) · Level 3
Find \(f'(a)\), where \(a\) is in the domain of \(f\): \(f(x) = \sqrt{x - 2}\).
25 Derivative and Tangent Lines · Level 3
Let \(f(x) = x^3 - 2 x + 4\).
(a) Find \(f'(a)\).
(b) Find equations of the tangent lines to the graph of \(f\) at the points whose \(x\)-coordinates are 0, 1, and 2.
(c) Graph \(f\) and the three tangent lines.

Enter your answer directly below each part above.

26 Derivative and Tangent Lines · Level 3
Let \(g(x) = \dfrac{1}{2 x - 1}\).
(a) Find \(g'(a)\).
(b) Find equations of the tangent lines to the graph of \(g\) at the points whose \(x\)-coordinates are \(-1\), 0, and 1.
(c) Graph \(g\) and the three tangent lines.

Enter your answer directly below each part above.

27 Application - Velocity of a Ball · Level 3
Velocity of a Ball: If a ball is thrown straight up with a velocity of 40 ft/s, its height (in feet) after \(t\) seconds is given by \(y = 40 t - 16 t^2\). Find the instantaneous velocity when \(t = 2\).
28 Application - Velocity on the Moon · Level 4
Velocity on the Moon: If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters) after \(t\) seconds is given by \(H = 58 t - 0.83 t^2\).
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(a) Find the instantaneous velocity of the arrow after one second.
(b) Find the instantaneous velocity of the arrow when \(t = a\).
(c) At what time \(t\) will the arrow hit the moon?
(d) With what velocity will the arrow hit the moon?

Enter your answer directly below each part above.

29 Application - Velocity of a Particle · Level 3
Velocity of a Particle: The displacement \(s\) (in meters) of a particle moving in a straight line is given by the equation of motion \(s = 4 t^3 + 6 t + 2\), where \(t\) is measured in seconds. Find the instantaneous velocity of the particle at times \(t = a\), \(t = 1\), \(t = 2\), and \(t = 3\).
30 Application - Inflating a Balloon · Level 3
Inflating a Balloon: A spherical balloon is being inflated. Find the rate of change of the surface area \((S = 4 \pi r^2)\) with respect to the radius \(r\) when \(r = 2\) ft.
31 Exercise - Rate of Change (Heart Rate) · Level 3
*Heart Rate.* 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. Data: at \(t = 36, 38, 40, 42, 44\) minutes, the heartbeats are \(2530, 2661, 2806, 2948, 3080\) respectively. *(a)* Find the average heart rates (slopes of the secant lines) over the time intervals \([40, 42]\) and \([42, 44]\). *(b)* Estimate the patient's heart rate after 42 minutes by averaging the slopes of these two secant lines.
32 Exercise - Rate of Change (Water Flow) · Level 3
*Water Flow.* 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. Data: at \(t = 5, 10, 15, 20, 25, 30\) minutes, \(V = 694, 444, 250, 111, 28, 0\) gallons respectively. *(a)* Find the average rates at which water flows from the tank (slopes of secant lines) for the time intervals \([10, 15]\) and \([15, 20]\). *(b)* The slope of the tangent line at the point \((15, 250)\) represents the rate at which water is flowing from the tank after 15 minutes. Estimate this rate by averaging the slopes of the secant lines in part (a).
33 Exercise - Rate of Change (World Population) · Level 3
*World Population Growth.* The table gives the world's population in the 20th century. Data (Year: Population in millions): - 1900: 1650 - 1910: 1750 - 1920: 1860 - 1930: 2070 - 1940: 2300 - 1950: 2560 - 1960: 3040 - 1970: 3710 - 1980: 4450 - 1990: 5280 - 2000: 6080 Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.
34 Discovery - Estimating Derivatives from a Graph · Level 3
*Estimating Derivatives from a Graph.* For the function \(g\) whose graph is given, arrange the following numbers in increasing order, and explain your reasoning: \(g'(-2), g'(0), g'(2), g'(4)\)
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35 Discovery - Estimating Velocities from a Graph · Level 3
*Estimating Velocities from a Graph.* The graph shows the position function of a car. Use the shape of the graph to explain your answers to the following questions. *(a)* What was the initial velocity of the car? *(b)* Was the car going faster at \(B\) or at \(C\)? *(c)* Was the car slowing down or speeding up at \(A\), \(B\), and \(C\)? *(d)* What happened between \(D\) and \(E\)?
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36 Example - Tangent Line to a Hyperbola · Level 3
Find an equation of the tangent line to the hyperbola \(y = \dfrac{3}{x}\) at the point \((3, 1)\).
37 Example - Finding a Tangent Line · Level 2
Find an equation of the tangent line to the curve \(y = x^3 - 2 x + 3\) at the point \((1, 2)\).
38 Example - Finding a Derivative at a Point · Level 2
Find the derivative of the function \(f(x) = 5 x^2 + 3 x - 1\) at the number 2.
39 Example - Finding a Derivative · Level 3
Let \(f(x) = \sqrt{x}\).
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(a) Find \(f'(a)\).
(b) Find \(f'(1)\), \(f'(4)\), and \(f'(9)\).

Enter your answer directly below each part above.

40 Example - Instantaneous Velocity · Level 3
If an object is dropped from a height of 3000 ft, its distance above the ground (in feet) after \(t\) seconds is given by \(h(t) = 3000 - 16 t^2\). Find the object's instantaneous velocity after 4 seconds.
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41 Example - Estimating an Instantaneous Rate of Change · Level 3
Let \(P(t)\) be the population of the United States at time \(t\). Midyear population estimates are: \(P(1996) = 269,667,000\); \(P(1998) = 276,115,000\); \(P(2000) = 282,192,000\); \(P(2002) = 287,941,000\); \(P(2004) = 293,655,000\). Interpret and estimate the value of \(P'(2000)\).

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