In Exercises 99 and 100, use the position function for free-falling objects. 99. A

In Exercises 1 and 2, use the graph to estimate the slope of the tangent line to \(y=x^{n}\) at the point (1, 1). Verify your answer analytically. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) \(y=x^{1 / 2}\) (b) \(y=x^{3}\) Text Transcription: y = x^n y = x^1/2 y = x^3

In Exercises 1 and 2, use the graph to estimate the slope of the tangent line to \(y=x^{n}\) at the point (1, 1). Verify your answer analytically. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) \(y=x^{-1 / 2}\) (b) \(y=x^{-1}\) Text Transcription: y = x^n y = x^-1/2 y = x^-1

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. y = 12

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. f(x) = - 9

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=x^{7}\) Text Transcription: y = x^7

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=x^{16}\) Text Transcription: y = x^16

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=\frac{1}{x^{5}}\) Text Transcription: y = 1 / x^5

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=\frac{1}{x^{8}}\) Text Transcription: y = 1 / x^8

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(f(x)=\sqrt[5]{x}\) Text Transcription: f(x) = 5th root x

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(g(x)=\sqrt[4]{x}\) Text Transcription: g(x) = 4th root x

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. f(x) = x + 11

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. g(x) = 3x - 1

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(f(t)=-2 t^{2}+3 t-6\) Text Transcription: f(t) = - 2t^{2} + 3t - 6

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=t^{2}+2 t-3\) Text Transcription: y = t^2 + 2t - 3

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(g(x)=x^{2}+4 x^{3}\) Text Transcription: g(x) = x^2 + 4x^{3}

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=8-x^{3}\) Text Transcription: y = 8 - x^3

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(s(t)=t^{3}+5 t^{2}-3 t+8\) Text Transcription: s(t) = t^3 + 5 t^2 - 3t + 8

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(f(x)=2 x^{3}-x^{2}+3 x\) Text Transcription: f(x) = 2x^3 - x^2 + 3x

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=\frac{\pi}{2} \sin \theta-\cos \theta\) Text Transcription: y = pi / 2 sin theta - cos theta

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(g(t)=\pi \cos t\) Text Transcription: g(t) = pi cos t

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=x^{2}-\frac{1}{2} \cos x\) Text Transcription: y = x^2 - 1/2 cos x

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. y = 7 + sin x

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=\frac{1}{x}-3 \sin x\) Text Transcription: y = 1 / x - 3 sin x

In Exercises 3 –24, use the rules of differentiation to find the derivative of the function. \(y=\frac{5}{(2 x)^{3}}+2 \cos x\) Text Transcription: y = 5 / (2x)^3 + 2 cos x

In Exercises 25-30, complete the table. Original Function Rewrite Differentiate Simplify \(y=\frac{5}{2 x^{2}}\) ______ _________ _______ Text Transcription: y = 5/2x^2

In Exercises 25-30, complete the table. Original Function Rewrite Differentiate Simplify \(y=\frac{2}{3 x^{2}}\) ______ _________ _______ Text Transcription: y = 2/3x^2

In Exercises 25-30, complete the table. Original Function Rewrite Differentiate Simplify \(y=\frac{6}{(5 x)^{3}}\) ______ _________ _______ Text Transcription: y = 6 / (5 x)^3

In Exercises 25-30, complete the table. Original Function Rewrite Differentiate Simplify \(y=\frac{\pi}{(3 x)^{2}}\) ______ _________ _______ Text Transcription: y = pi/(3x)^2

In Exercises 25-30, complete the table. Original Function Rewrite Differentiate Simplify \(y=\frac{\sqrt{x}}{x}\) ______ _________ _______ Text Transcription: y = sqrt x/ x

In Exercises 25-30, complete the table. Original Function Rewrite Differentiate Simplify \(y=\frac{4}{x^{-3}}\) ______ _________ _______ Text Transcription: y = 4 / x^-3

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(f(x)=\frac{8}{x^{2}}\) (2, 2) Text Transcription: f(x) = 8 / x^2

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(f(t)=3-\frac{3}{5 t}\) \(\left(\frac{3}{5}, 2\right)\) Text Transcription: f(t) = 3 - 3/5t (3/5, 2)

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(f(x)=-\frac{1}{2}+\frac{7}{5} x^{3}\) \(\left(0,-\frac{1}{2}\right)\) Text Transcription: f(x) = - 1/2 + 7/5 x^3 (0, -1/2)

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(y=3 x^{3}-10\) (2, 14) Text Transcription: y = 3x^3 -10

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(y=(4 x+1)^{2}\) (0, 1) Text Transcription: y = (4x + 1)^2

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(f(x)=3(5-x)^{2}\) (5, 0) Text Transcription: f(x) = 3(5 - x)^2

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point \(f(\theta)=4 \sin \theta-\theta\) (0, 0) Text Transcription: f(theta) = 4 sin theta -theta

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function Point g(t) = –2 cos t + 5 \((\pi, 7)\) Text Transcription: (pi, 7)

In Exercises 39–54, find the derivative of the function. \(f(x)=x^2+5-3x^{-2}\) Text Transcription: f(x)=x^2+5-3x^-2

In Exercises 39–54, find the derivative of the function. \(f(x)=x^2-3x-3x^{-2}\) Text Transcription: f(x)=x^2-3x-3x^-2

In Exercises 39–54, find the derivative of the function. \(g(t)=t^2-\frac{4}{t^3}\) Text Transcription: g(t)=t^2 -4/t^3

In Exercises 39–54, find the derivative of the function. \(f(x)=x+\frac{1}{x^2}\) Text Transcription: f(x)=x+ 1/x^2

In Exercises 39–54, find the derivative of the function. \(f(x)=\frac{4x^3+3x^2}{x}\) Text Transcription: f(x)=4x^3 +3x^2 /x

In Exercises 39–54, find the derivative of the function. \(f(x)=\frac{x^3-6}{x^2}\) Text Transcription: f(x)=x^3 -6 /x^2

In Exercises 39–54, find the derivative of the function. \(f(x)=\frac{x^3-3x^2+4}{x^2}\) Text Transcription: f(x)=x^3 -3x^2 +4 /x^2

In Exercises 39–54, find the derivative of the function. \(h(x)=\frac{2x^2-3x+1}{x}\) Text Transcription: h(x)=2x^2 -3x+1 /x

In Exercises 39–54, find the derivative of the function. \(y=x(x^2+1)\) Text Transcription: y=x(x^2 +1)

In Exercises 39–54, find the derivative of the function. \(y=3x(6x-5x^2)\) Text Transcription: y=3x(6x-5x^2)

In Exercises 39–54, find the derivative of the function. \(f(x)=\sqrt{x}-6\sqrt[3]{x}\) Text Transcription: f(x)=sqrtx - 6 cube rootx

In Exercises 39–54, find the derivative of the function. \(f(x)=\sqrt[3]{x}+\sqrt[5]{x}\) Text Transcription: f(x)=cube rootx + 5th rootx

In Exercises 39–54, find the derivative of the function. \(h(s)=s^{4/5}-s^{2/3}\) Text Transcription: h(s)=s^4/5 - s^2/3

In Exercises 39–54, find the derivative of the function. \(f(t)=t^{2/3}-t^{1/3}+4\) Text Transcription: f(t)=t^2/3 -t^1/3 +4

In Exercises 39–54, find the derivative of the function. \(f(x)=6\sqrt{x}+5 \cos x\) Text Transcription: f(x)=6sqrtx +5 cos x

In Exercises 39–54, find the derivative of the function. \(f(x)=\frac{2}{\sqrt[3]{x}}+3 \cos x\) Text Transcription: f(x)=2/cube rootx +3 cos x

In Exercises 55–58, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. Function Point \(y=x^4-3x^2+2\) (1, 0) Text Transcription: y=x^4 -3x^2 +2

In Exercises 55–58, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. Function Point \(y=x^3+x\) (-1, -2) Text Transcription: y=x^3 +x

In Exercises 55–58, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. Function Point \(f(x)=\frac{2}{\sqrt[4]{x^3}}\) (1, 2) Text Transcription: f(x)=2/4th rootx^3

In Exercises 55–58, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. Function Point \(y=(x^2=2x)(x+1)\) (1, 6) Text Transcription: y=(x^2 +2x)(x+1)

In Exercises 59–64, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=x^4-2x^2+3\) Text Transcription: y=x^4 -2x^2 +3

In Exercises 59–64, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=x^3+x\) Text Transcription: y=x^3 +x

In Exercises 59–64, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=\frac{1}{x^2}\) Text Transcription: y=1/x^2

In Exercises 59–64, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=x^2+9\) Text Transcription: y=x^2 +9

In Exercises 59–64, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=x+ \sin x,\ \ \ 0\ \leq\ x\ <\ 2\pi\) Text Transcription: y=x + sin x, 0 leq x < 2pi

In Exercises 59–64, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=\sqrt{3x}+2 \cos x,\ \ \ 0\ \leq\ x\ <\ 2\pi\)

In Exercises 65–70, find k such that the line is tangent to the graph of the function. Function Line \(f(x)=x^2-kx\) \(y=5x+4\) Text Transcription: f(x)=x^2 -kx y=5x-4

In Exercises 65–70, find k such that the line is tangent to the graph of the function. Function Line \(f(x)=k-x^2\) \(y=-6x+1\) Text Transcription: f(x)=k-x^2 y=-6x+1

In Exercises 65–70, find k such that the line is tangent to the graph of the function. Function Line \(f(x)=\frac{k}{x}\) \(y=-\frac{3}{4}x=3\) Text Transcription: f(x)=k/x y=-3/4 x+3

In Exercises 65–70, find k such that the line is tangent to the graph of the function. Function Line \(f(x)=k\sqrt{x}\) \(y=x+4\) Text Transcription: f(x)=ksqrtx y=x+4

In Exercises 65–70, find k such that the line is tangent to the graph of the function. Function Line \(f(x)=kx^3\) \(y=x+1\) Text Transcription: f(x)=kx^3 y=x+1

In Exercises 65–70, find k such that the line is tangent to the graph of the function. Function Line \(f(x)=kx^4\) \(y=4x-1\) Text Transcription: f(x)=kx^4 y=4x-1

Sketch the graph of a function f such that f’ > 0 for all x and the rate of change of the function is decreasing.

Use the graph of f to answer each question. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) Between which two consecutive points is the average rate of change of the function greatest? (b) Is the average rate of change of the function between A and B greater than or less than the instantaneous rate of change at B? (c) Sketch a tangent line to the graph between C and D such that the slope of the tangent line is the same as the average rate of change of the function between C and D.

In Exercises 73 and 74, the relationship between f and g is given. Explain the relationship between f’ and g’. g(x) = f(x) + 6

In Exercises 73 and 74, the relationship between f and g is given. Explain the relationship between f’ and g’. g(x) = -f(x)

In Exercises 75 and 76, the graphs of a function f and its derivative f’ are shown on the same set of coordinate axes. Label the graphs as f or f’ and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 75 and 76, the graphs of a function f and its derivative f’ are shown on the same set of coordinate axes. Label the graphs as f or f’ and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

Sketch the graphs of \(y=x^2\) and \(y=-x^2+6x-5\), and sketch the two lines that are tangent to both graphs. Find equations of these lines. Text Transcription: y=x^2 y=-x^2 +6x-5

Show that the graphs of the two equations y = x and y = 1/x have tangent lines that are perpendicular to each other at their point of intersection.

Show that the graph of the function f(x) = 3x + sin x + 2 does not have a horizontal tangent line.

Show that the graph of the function \(f(x)=x^5+3x^3+5x\) does not have a tangent line with a slope of 3.

In Exercise 81 and 82, find an equation of the tangent line to the graph of the function f through the point \((x_0,\ y_0)\) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation \(f'(x)=\frac{y_0-y}{x_0-x}\). \(f(x)=\sqrt{x}\) \((x_0,\ y_0)\ =\ (-4,\ 0)\) Text Transcription: f’(x)=y_0 -y/ x_0 -x f(x)=sqrtx (x_0, y_0) = (-4, 0)

In Exercise 81 and 82, find an equation of the tangent line to the graph of the function f through the point \((x_0,\ y_0)\) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation \(f'(x)=\frac{y_0-y}{x_0-x}\). \(f(x)=\frac{2}{x}\) \((x_0,\ y_0)\ =\ (5,\ 0)\) Text Transcription: f’(x)=y_0 -y/ x_0 -x f(x)=2/x (x_0, y_0) = (5, 0)

Linear Approximation Use a graphing utility, with a square window setting, to zoom in on the graph of \(f(x)=4-\frac{1}{2}x^2\) to approximate f’(1). Use the derivative to find f’(1). Text Transcription: f(x)=4-1/2 x^2

Linear Approximation Use a graphing utility, with a square window setting, to zoom in on the graph of \(f(x)=4\sqrt{x}+1\) to approximate f’(4). Use the derivative to find f’(4). Text Transcription: f(x)=4sqrtx +1

Linear Approximation Consider the function \(f(x)=x^{3/2}\) with the solution point (4, 8). (a) Use a graphing utility to graph f. Use the zoom feature to obtain successive magnifications of the graph in the neighborhood of the point (4, 8). After zooming in a few times, the graph should appear nearly linear. Use the trace feature to determine the coordinates of a point near (4, 8). Find an equation of the secant line S(x) through the two points. (b) Find the equation of the line T(x) = f’(4)(x - 4) + f(4) tangent to the graph of f passing through the given point. Why are the linear functions S and T nearly the same? (c) Use a graphing utility to graph f and T on the same set of coordinate axes. Note that T is a good approximation of f when x is close to 4. What happens to the accuracy of the approximation as you move farther away from the point of tangency? (d) Demonstrate the conclusion in part (c) by completing the table. Text Transcription: f(x)=x^3/2

Linear Approximation Repeat Exercise 85 for the function \(f(x) = x^3\) where T(x) is the line tangent to the graph at the point (1, 1). Explain why the accuracy of the linear approximation decreases more rapidly than in Exercise 85. Text Transcription: \(f(x)=x^3\)

True or False? In Exercises 87–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f’(x) = g’(x), then f(x) = g(x).

True or False? In Exercises 87–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) = g(x) + c, then f’(x) = g’(x).

True or False? In Exercises 87–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=\pi^2\), then \(dy/dx=2\pi\). Text Transcription: y=pi^2 dy/dx = 2pi

True or False? In Exercises 87–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=x/\pi\), then \(dy/dx=1/\pi\). Text Transcription: y=x/pi dy/dx=1/pi

True or False? In Exercises 87–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If g(x) = 3f(x), then g’(x) = 3f’(x).

True or False? In Exercises 87–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=1/x^n\), then \(f'(x)=1/(nx^{n-1})\). Text Transcription: f(x)=1/x^n f’(x)=1/(nx^n-1)

In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(t) = 4t + 5, [1, 2]

In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. \(f(t)=t^2-7,\ \ \ [3,\ 3.1]\) Text Transcription: f(t)=t^2 -7, [3, 3.1]

In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. \(f(x)=\frac{-1}{x},\ \ \ [1,\ 2]\) Text Transcription: f(x)=-1/x , [1, 2]

In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. \(f(x)=\sin x,\ \ \ [0,\ \frac{\pi}{6}]\) Text Transcription: f(x)=sin x, [0, \pi/6]

Vertical Motion In Exercises 97 and 98, use the position function \(s(t)=-16t^2+v_0+s_0\) for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval [1, 2]. (c) Find the instantaneous velocities when t = 1 and t = 2. (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact. Text Transcription: s(t)=-16t^2 +v_0 + s_0

Vertical Motion In Exercises 97 and 98, use the position function \(s(t)=-16t^2+v_0+s_0\) for free-falling objects. A ball is thrown straight down from the top of a 220-foot building with an initial velocity of -22 feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet? Text Transcription: s(t)=-16t^2 +v_0 + s_0

Vertical Motion In Exercises 99 and 100, use the position function \(s(t)=-4.9t^2+v_0+s_0\) for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds? Text Transcription: s(t)=-4.9t^2 +v_0 + s_0

Vertical Motion In Exercises 99 and 100, use the position function \(s(t)=-4.9t^2+v_0+s_0\) for free-falling objects. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 5.6 seconds after the stone is dropped? Text Transcription: s(t)=-4.9t^2 +v_0 + s_0

Think About It In Exercises 101 and 102, the graph of a position function is shown. It represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function.

Think About It In Exercises 101 and 102, the graph of a position function is shown. It represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function.

Think About It In Exercises 103 and 104, the graph of a velocity function is shown. It represents the velocity in miles per hour during a 10-minute drive to work. Make a sketch of the corresponding position function.

Think About It In Exercises 103 and 104, the graph of a velocity function is shown. It represents the velocity in miles per hour during a 10-minute drive to work. Make a sketch of the corresponding position function.

Modeling Data The stopping distance of an automobile, on dry, level pavement, traveling at a speed (kilometers per hour) is the distance (meters) the car travels during the reaction time of the driver plus the distance (meters) the car travels after the brakes are applied (see figure). The table shows the results of an experiment. (a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance. (b) Use the regression capabilities of a graphing utility to find a quadratic model for braking distance. (c) Determine the polynomial giving the total stopping distance T. (d) Use a graphing utility to graph the functions R, B, and T in the same viewing window. (e) Find the derivative of T and the rates of change of the total stopping distance for v = 40, v = 80, and v = 100. (f ) Use the results of this exercise to draw conclusions about the total stopping distance as speed increases.

Fuel Cost A car is driven 15,000 miles a year and gets x miles per gallon. Assume that the average fuel cost is $2.76 per gallon. Find the annual cost of fuel C as a function of x and use this function to complete the table. Who would benefit more from a one-mile-per-gallon increase in fuel efficiency—the driver of a car that gets 15 miles per gallon or the driver of a car that gets 35 miles per gallon? Explain.

Volume The volume of a cube with sides of length is given by \(V = s^3\). Find the rate of change of the volume with respect to s when s = 6 centimeters. Text Transcription: V=s^3

Area The area of a square with sides of length is given by \(A = s^2\). Find the rate of change of the area with respect to s when s = 6 meters.

Velocity Verify that the average velocity over the time interval \([t_0-\triangle t,\ t_0+\triangle t]\) is the same as the instantaneous velocity at \(t = t_0\) for the position function \(s(t)=-\frac{1}{2}at^2+c\) Text Transcription: [t_0 - delta t, t_0 + delta t] t=t_0 s(t)=-1/2 at^2 +c

Inventory Management The annual inventory cost C for a manufacturer is \(C=\frac{1,008,000}{Q}+6.3Q\) where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 350 to 351, and compare this with the instantaneous rate of change when Q = 350. Text Transcription: C=1,008,000/Q + 6.3Q

Writing The number of gallons N of regular unleaded gasoline sold by a gasoline station at a price of p dollars per gallon is given by N = f(p). (a) Describe the meaning of f’(2.979). (b) Is f’(2.979) usually positive or negative? Explain.

Newton’s Law of Cooling This law states that the rate of change of the temperature of an object is proportional to the difference between the object’s temperature T and the temperature \(T_a\) of the surrounding medium. Write an equation for this law. Text Transcription: T_a

Find an equation of the parabola \(y=ax^2+bx+c\) that passes through (0, 1) and is tangent to the line y = x - 1 at (1, 0). Text Transcription: y=ax^2 +bx + c

Let (a, b) be an arbitrary point on the graph of y = 1/x, x > 0. Prove that the area of the triangle formed by the tangent line through (a, b) and the coordinate axes is 2.

Find the tangent line(s) to the curve \(y = x^3 - 9x\) through the point (1, -9). Text Transcription: y=x^3 -9x

Find the equation(s) of the tangent line(s) to the parabola \(y = x^2\) through the given point. (a) (0, a) (b) (a, 0) Are there any restrictions on the constant a? Text Transcription: y=x^2

In Exercises 117 and 118, find a and b such that f is differentiable everywhere. \(f(x)=\left\{\begin{array}{ll} a x^{3}, & x \leq 2 \\ x^{2}+b, & x>2 \end{array}\right.\) Text Transcription: f(x)={ax^3, x leq 2 _ x^2 + b, x > 2

In Exercises 117 and 118, find a and b such that f is differentiable everywhere. \(f(x)=\left\{\begin{array}{ll} \cos x, & x<0 \\ a x+b, & x \geq 0 \end{array}\right.\) Text Transcription: f(x)={cos x, x < 0 _ ax + b, x geq 0

Where are the functions \(f_1(x)=|\sin x|\ \text{and}\ f_2(x)=\sin |x|\) differentiable? Text Transcription: f_1 (x) = | sin x | and f_2 (x) = sin |x|

Prove that \(\frac{d}{dx}\ [\cos x]=-\sin x\). Text Transcription: d/dx [cos x] = -sin x