In Exercises 70 75, graphs of functions are shown in by

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 1 14, a. Find the slope of the tangent line to the graph of at the given point. b. Find the slope-intercept equation of the tangent line to the graph of at the given point.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 15 28, a. Find the derivative of at That is, find b. Find the slope of the tangent line to the graph of at each of the two values of given to the right of the function.

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

In Exercises 29 36, a. Use transformations of a common graph to obtain the graph of b. Find the slope-intercept equation of the tangent line to the graph of at the point whose is given. c. Use the and the slope to graph the tangent line in the same rectangular coordinate system as the graph of

The function describes the area of a square, in square inches, whose sides each measure inches. If is changing, a. Find the average rate of change of the area with respect to as changes from 6 inches to 6.1 inches and from 6 inches to 6.01 inches. b. Find the instantaneous rate of change of the area with respect to at the moment when inches

The function describes the area of a square, in square inches, whose sides each measure inches. If is changing, a. Find the average rate of change of the area with respect to as changes from 10 inches to 10.1 inches and from 10 inches to 10.01 inches. b. Find the instantaneous rate of change of the area with respect to at the moment when inches.

The function describes the area of a circle, in square inches, whose radius measures inches. If the radius is changing, a. Find the average rate of change of the area with respect to the radius as the radius changes from 2 inches to 2.1 inches and from 2 inches to 2.01 inches. b. Find the instantaneous rate of change of the area with respect to the radius when the radius is 2 inches.

The function describes the area of a circle, in square inches, whose radius measures inches. If the radius is changing, a. Find the average rate of change of the area with respect to the radius as the radius changes from 4 inches to 4.1 inches and from 4 inches to 4.01 inches. b. Find the instantaneous rate of change of the area with respect to the radius when the radius is 4 inches.

The function describes the surface area, of a sphere of radius inches. If the radius is changing, find the instantaneous rate of change of the surface area with respect to the radius when the radius is 6 inches

The function describes the volume, of a right circular cylinder of height 5 feet and radius feet. If the radius is changing, find the instantaneous rate of change of the volume with respect to the radius when the radius is 8 feet.

An explosion causes debris to rise vertically with an initial velocity of 64 feet per second. The function describes the height of the debris above the ground, in feet, seconds after the explosion. a. What is the instantaneous velocity of the debris 1 second after the explosion? 3 seconds after the explosion? b. What is the instantaneous velocity of the debris when it hits the ground?

An explosion causes debris to rise vertically with an initial velocity of 72 feet per second. The function describes the height of the debris above the ground, in feet, seconds after the explosion.

A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 96 feet per second.The function describes the ball s height above the ground, in feet, seconds after it is hit. a. What is the instantaneous velocity of the ball 2 seconds after it is hit? 4 seconds after it is hit? b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? What is its maximum height?

A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 64 feet per second.The function describes the ball s height above the ground, in feet, seconds after it is hit. a. What is the instantaneous velocity of the ball 1 second after it is hit? 3 seconds after it is hit? b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? What is its maximum height?

Explain how the tangent line to the graph of a function at point is related to the secant lines between points and on the function s graph.

Explain what we mean by the slope of the graph of a function at a point.

Explain how to find the slope of f1x2 = x

Explain how to write an equation of the tangent line to the graph of

If you are given the equation of function describe how to find

Explain how to use the derivative to compute the slopes of various tangent lines to the graph of a function.

Explain how the instantaneous rate of change of a function at a point is related to its average rates of change.

If a function expresses an object s position in terms of time, how do you find the instantaneous velocity of the object at any time during its motion?

Use the concept of an interval of time to describe how calculus views a particular instant of time.

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. Your friend quips, Well at least you know calculus; you can figure out the velocity with which the camera is going to hit the water. If the camera s height, in feet, over the water after seconds is describe how to determine the camera s velocity at the instant of its demise.

A calculus professor introduced the derivative by saying that it could be summed up in one word: slope. Explain what this means

For an unusual introduction to calculus by a poetic, quirky, and funny writer who loves the subject, read A Tour of the Calculus by David Berlinski (Vintage Books, 1995). Write a report describing two new things that you learned from the book about algebra, trigonometry, limits, or derivatives.

Use the feature of a graphing utility to graph the functions and tangent lines for any five exercises from Exercises 1 14. Use the equation that is displayed on the screen to verify the slope-intercept equation of the tangent line that you found in each exercise

Without using the feature of a graphing utility, graph the function and the tangent line whose equation you found for any five exercises from Exercises 1 14. Does the line appear to be tangent to the graph of at the point on that is given in the exercise?

Use the feature on a graphing utility that gives the derivative of a function evaluated at any number to verify part (b) for any five of your answers in Exercises 15 28

In Exercises 62 65, find, or approximate to two decimal places, the derivative of each function at the given number using a graphing utility

In Exercises 62 65, find, or approximate to two decimal places, the derivative of each function at the given number using a graphing utility

In Exercises 62 65, find, or approximate to two decimal places, the derivative of each function at the given number using a graphing utility

In Exercises 62 65, find, or approximate to two decimal places, the derivative of each function at the given number using a graphing utility

In Exercises 66 69, determine whether each statement makes sense or does not make sense, and explain your reasoning

In Exercises 66 69, determine whether each statement makes sense or does not make sense, and explain your reasoning

In Exercises 66 69, determine whether each statement makes sense or does not make sense, and explain your reasoning

In Exercises 66 69, determine whether each statement makes sense or does not make sense, and explain your reasoning

In Exercises 70 75, graphs of functions are shown in by viewing rectangles. Match each function with the graph of its derivative. Graphs of derivatives are labeled and are shown in by viewing rectangles

In Exercises 70 75, graphs of functions are shown in by viewing rectangles. Match each function with the graph of its derivative. Graphs of derivatives are labeled and are shown in by viewing rectangles

In Exercises 70 75, graphs of functions are shown in by viewing rectangles. Match each function with the graph of its derivative. Graphs of derivatives are labeled and are shown in by viewing rectangles

In Exercises 70 75, graphs of functions are shown in by viewing rectangles. Match each function with the graph of its derivative. Graphs of derivatives are labeled and are shown in by viewing rectangles

In Exercises 70 75, graphs of functions are shown in by viewing rectangles. Match each function with the graph of its derivative. Graphs of derivatives are labeled and are shown in by viewing rectangles

In Exercises 70 75, graphs of functions are shown in by viewing rectangles. Match each function with the graph of its derivative. Graphs of derivatives are labeled and are shown in by viewing rectangles

A ball is thrown straight up from a rooftop 96 feet high with an initial velocity of 80 feet per second. The function describes the ball s height above the ground, in feet, seconds after it is thrown. The ball misses the rooftop on its way down and eventually strikes the ground. What is its instantaneous velocity as it passes the rooftop on the way down?

Show that the rate of change of the area of a circle with respect to its radius is equal to the circumference of the circle

Show that the of the vertex of the parabola whose equation is occurs when the derivative of the function is zero.

For any positive integer prove that if then f1x2 = nxn-1 .