The derivative of f is the spike function in Figure 2.36.What can you say about the

(a) Estimate f (2) using the values of f in the table. (b) For what values of x does f (x) appear to be positive? Negative?

Find approximate values for f (x) at each of the x-values given in the following table.

For Exercises 312, graph the derivative of the given functions. 3

For Exercises 312, graph the derivative of the given functions. 4

For Exercises 312, graph the derivative of the given functions. 5

For Exercises 312, graph the derivative of the given functions. 6

For Exercises 312, graph the derivative of the given functions. 7

For Exercises 312, graph the derivative of the given functions. 8

For Exercises 312, graph the derivative of the given functions. 9

For Exercises 312, graph the derivative of the given functions. 10

For Exercises 312, graph the derivative of the given functions. 11

For Exercises 312, graph the derivative of the given functions. 12

For Exercises 1318, sketch the graph of f(x), and use this graph to sketch the graph of f (x). f(x)=5x

For Exercises 1318, sketch the graph of f(x), and use this graph to sketch the graph of f (x). f(x) = x2

For Exercises 1318, sketch the graph of f(x), and use this graph to sketch the graph of f (x). f(x) = x(x 1)

For Exercises 1318, sketch the graph of f(x), and use this graph to sketch the graph of f (x). f(x) = ex

For Exercises 1318, sketch the graph of f(x), and use this graph to sketch the graph of f (x). f(x) = cos x

For Exercises 1318, sketch the graph of f(x), and use this graph to sketch the graph of f (x). f(x) = ln x

In Exercises 1920, find a formula for the derivative using the power rule. Confirm it using difference quotients. k(x)=1/x

In Exercises 1920, find a formula for the derivative using the power rule. Confirm it using difference quotients. l(x)=1/x2

Find a formula for the derivatives of the functions in Exercises 2122 using difference quotients. g(x)=2x2 3

Find a formula for the derivatives of the functions in Exercises 2122 using difference quotients. m(x)=1/(x + 1)

In each case, graph a smooth curve whose slope meets the condition. (a) Everywhere positive and increasing gradually. (b) Everywhere positive and decreasing gradually. (c) Everywhere negative and increasing gradually (becoming less negative). (d) Everywhere negative and decreasing gradually (becoming more negative).

Draw a possible graph of y = f(x) given the following information about its derivative. f (x) > 0 for x < 1 f (x) < 0 for x > 1 f (x)=0 at x = 1

For f(x) = ln x, construct tables, rounded to four decimals, near x = 1, x = 2, x = 5, and x = 10. Use the tables to estimate f (1), f (2), f (5), and f (10). Then guess a general formula for f (x).

Given the numerical values shown, find approximate values for the derivative of f(x) at each of the x-values given. Where is the rate of change of f(x) positive? Where is it negative? Where does the rate of change of f(x) seem to be greatest?

Values of x and g(x) are given in the table. For what value of x does g (x) appear to be closest to 3?

In the graph of f in Figure 2.31, at which of the labeled x-values is (a) f(x) greatest? (b) f(x) least? (c) f (x) greatest? (d) f (x) least?

For Problems 2938, sketch the graph of f (x). 29

For Problems 2938, sketch the graph of f (x). 30

For Problems 2938, sketch the graph of f (x). 31

For Problems 2938, sketch the graph of f (x). 32

For Problems 2938, sketch the graph of f (x). 33

For Problems 2938, sketch the graph of f (x). 34

For Problems 2938, sketch the graph of f (x). 35

For Problems 2938, sketch the graph of f (x). 36

For Problems 2938, sketch the graph of f (x). 37

For Problems 2938, sketch the graph of f (x). 38

Roughly sketch the shape of the graph of a quadratic polynomial, f, if it is known that: (1, 3) is on the graph of f. f (0) = 3, f (2) = 1, f (3) = 0.

A vehicle moving along a straight road has distance f(t) from its starting point at time t. Which of the graphs in Figure 2.32 could be f (t) for the following scenarios? (Assume the scales on the vertical axes are all the same.) (a) A bus on a popular route, with no traffic (b) A car with no traffic and all green lights (c) A car in heavy traffic conditions

A child inflates a balloon, admires it for a while and then lets the air out at a constant rate. If V (t) gives the volume of the balloon at time t, then Figure 2.33 shows V  (t) as a function of t. At what time does the child: (a) Begin to inflate the balloon? (b) Finish inflating the balloon? (c) Begin to let the air out? (d) What would the graph of V  (t) look like if the child had alternated between pinching and releasing the open end of the balloon, instead of letting the air out at a constant rate?

Figure 2.34 shows a graph of voltage across an electrical capacitor as a function of time. The current is proportional to the derivative of the voltage; the constant of proportionality is positive. Sketch a graph of the current as a function of time.

Figure 2.35 is the graph of f , the derivative of a function f. On what interval(s) is the function f (a) Increasing? (b) Decreasing?

The derivative of f is the spike function in Figure 2.36. What can you say about the graph of f?

The population of a herd of deer is modeled by P(t) = 4000 + 500 sin  2t 2  where t is measured in years from January 1. (a) How does this population vary with time? Sketch a graph of P(t) for one year. (b) Use the graph to decide when in the year the population is a maximum. What is that maximum? Is there a minimum? If so, when? (c) Use the graph to decide when the population is growing fastest. When is it decreasing fastest? (d) Estimate roughly how fast the population is changing on the first of July.

The graph in Figure 2.37 shows the accumulated federal debt since 1970. Sketch the derivative of this function. What does it represent?

Draw the graph of a continuous function y = f(x) that satisfies the following three conditions: f (x) > 0 for x < 2, f (x) < 0 for 2 <x< 2, f (x)=0 for x > 2.

Draw the graph of a continuous function y = f(x) that satisfies the following three conditions: f (x) > 0 for 1 <x< 3 f (x) < 0 for x < 1 and x > 3 f (x)=0 at x = 1 and x = 3

If lim x f(x) = 50 and f (x) is positive for all x, what is lim x f (x)? (Assume this limit exists.) Explain your answer with a picture.

Using a graph, explain why if f(x) is an even function, then f (x) is odd.

Using a graph, explain why if g(x) is an odd function, then g (x) is even.

In Problems 5254, explain what is wrong with the statement. The graph of the derivative of the function f(x) = cos x is always above the x-axis

In Problems 5254, explain what is wrong with the statement. A function, f, whose graph is above the x-axis for all x has a positive derivative for all x.

In Problems 5254, explain what is wrong with the statement. If f (x) = g (x) then f(x) = g(x).

In Problems 5556, give an example of: A function representing the position of a particle which has positive velocity for 0 <t< 0.5 and negative velocity for 0.5 <t< 1.

In Problems 5556, give an example of: A family of linear functions all with the same derivative.

Are the statements in Problems 5760 true or false? Give an explanation for your answer. The derivative of a linear function is constant.

Are the statements in Problems 5760 true or false? Give an explanation for your answer. If g(x) is a vertical shift of f(x), then f (x) = g (x).

Are the statements in Problems 5760 true or false? Give an explanation for your answer. If f (x) is increasing, then f(x) is also increasing.

Are the statements in Problems 5760 true or false? Give an explanation for your answer. If f(a) = g(a), then f (a) = g (a).