Solution: 2131 Find the derivative of the function using the definition of a derivative

12 Use the given graph to estimate the value of each derivative. Then sketch the graph of f9. 1. (a) f9s23d (b) f9s22d (c) f9s21d (d) f9s0d (e) f9s1d (f) f9s2d (g) f9s3d y x 1 1

12 Use the given graph to estimate the value of each derivative. Then sketch the graph of f9. 2. (a) f9s0d (b) f9s1d (c) f9s2d (d) f9s3d (e) f9s4d (f) f9s5d (g) f9s6d (h) f9s7d

Match the graph of each function in (a)(d) with the graph of its derivative in IIV. Give reasons for your choices.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

411 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f9 below it.

Yeast population Shown is the graph of the population function Pstd for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative P9std. What does the graph of P9 tell us about the yeast population?

Tadpole weights The graph shows the average body weight W as a function of time for tadpoles raised in a density of 80 tadpolesyL. (a) What is the meaning of the derivative W9std? (b) Sketch the graph of W9std.

Ground reaction force in walking The graph shows the horizontal force Fstd exerted by the ground on a person who is walking. (a) What is the meaning of the derivative F9std? (b) Sketch the graph of F9std.

Marriage age The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function M9std. During which years was the derivative negative?

1618 Make a careful sketch of the graph of f and below it sketch the graph of f9 in the same manner as in Exercises 411. Can you guess a formula for f9sxd from its graph? 16. fsxd sin x 17. fsxd ex 18. fsxd ln x

1618 Make a careful sketch of the graph of f and below it sketch the graph of f9 in the same manner as in Exercises 411. Can you guess a formula for f9sxd from its graph? 16. fsxd sin x 17. fsxd ex 18. fsxd ln x

1618 Make a careful sketch of the graph of f and below it sketch the graph of f9 in the same manner as in Exercises 411. Can you guess a formula for f9sxd from its graph? 16. fsxd sin x 17. fsxd ex 18. fsxd ln x

Let fsxd x 2 . (a) Estimate the values of f9s0d, f9( 1 2), f9s1d, and f9s2d by using a graphing device to zoom in on the graph of f. (b) Use symmetry to deduce the values of f9(21 2 ), f9s21d, and f9s22d. (c) Use the results from parts (a) and (b) to guess a formula for f9sxd. (d) Use the definition of a derivative to prove that your guess in part (c) is correc

Let fsxd x 3 . (a) Estimate the values of f9s0d, f9( 1 2), f9s1d, f9s2d, and f9s3d by using a graphing device to zoom in on the graph of f. (b) Use symmetry to deduce the values of f9(21 2 ), f9s21d, f9s22d, and f9s23d. (c) Use the values from parts (a) and (b) to graph f9. (d) Guess a formula for f9sxd. (e) Use the definition of a derivative to prove that your guess in part (d) is correct.

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

2131 Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative. 21. fsxd 1 2 x 2 1 3 22. fsxd mx 1 b 23. fstd 5t 2 9t 2 24. fsxd 1.5x 2 2 x 1 3.7 25. fsxd x 2 2 2x 3 26. fsxd x 1 sx 27. tsxd s1 1 2x 28. f sxd x 2 2 1 2x 2 3 29. Gstd 4t t 1 1 30. tstd 1 st 31. fsxd x

3234 (a) Use the definition of the derivative to calculate f9. (b) Check to see that your answer is reasonable by comparing the graphs of f and f9. 32. fsxd x 1 1yx 33. fsxd x 4 1 2x 34. fstd t 2 2 st

3234 (a) Use the definition of the derivative to calculate f9. (b) Check to see that your answer is reasonable by comparing the graphs of f and f9. 32. fsxd x 1 1yx 33. fsxd x 4 1 2x 34. fstd t 2 2 st

3234 (a) Use the definition of the derivative to calculate f9. (b) Check to see that your answer is reasonable by comparing the graphs of f and f9. 32. fsxd x 1 1yx 33. fsxd x 4 1 2x 34. fstd t 2 2 st

Malarial parasites An experiment measured the number of malarial parasites Nstd per microliter of blood, where t is measured in days. The results of the experiment are shown in the table.

Blood alcohol concentration Researchers measured the blood alcohol concentration Cstd of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in mgymL) of the eight men. t (hours) 0.0 0.2 0.5 0.75 1.0 1.25 Cstd 0 0.25 0.41 0.40 0.33 0.29 t (hours) 1.5 1.75 2.0 2.25 2.5 3.0 Cstd 0.24 0.22 0.18 0.15 0.12 0.07 (a) What is the meaning of C9std? (b) Make a table of estimated values for C9std. Source: Adapted from P. Wilkinson et al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmaceutics, 5 (1977): 20724.

3740 The graph of f is given. State, with reasons, the numbers at which f is not differentiable

3740 The graph of f is given. State, with reasons, the numbers at which f is not differentiable

3740 The graph of f is given. State, with reasons, the numbers at which f is not differentiable

3740 The graph of f is given. State, with reasons, the numbers at which f is not differentiable

Graph the function fsxd x 1 s| x | . Zoom in repeatedly, first toward the point s21, 0d and then toward the origin. What is different about the behavior of f in the vicinity of these two points? What do you conclude about the differentiability of f ?

Zoom in toward the points (1, 0), (0, 1), and s21, 0d on the graph of the function tsxd sx 2 2 1d 2y3 . What do you notice? Account for what you see in terms of the differentiability of t

The figure shows the graphs of f, f9, and f 0. Identify each curve, and explain your choices.

The figure shows graphs of f, f9, f 0, and f -. Identify each curve, and explain your choices.

4546 Use the definition of a derivative to find f9sxd and f 0sxd. Then graph f, f9, and f 0 on a common screen and check to see if your answers are reasonable. 45. fsxd 3x 2 1 2x 1 1 46. fsxd x 3 2 3x

4546 Use the definition of a derivative to find f9sxd and f 0sxd. Then graph f, f9, and f 0 on a common screen and check to see if your answers are reasonable. 45. fsxd 3x 2 1 2x 1 1 46. fsxd x 3 2 3x

4748 The graph of the derivative f9 of a function f is shown. (a) On what intervals is f increasing? (b) If it is known that fs0d 0, sketch a possible graph of f

4748 The graph of the derivative f9 of a function f is shown. (a) On what intervals is f increasing? (b) If it is known that fs0d 0, sketch a possible graph of f

Recall that a function f is called even if fs2xd fsxd for all x in its domain and odd if fs2xd 2fsxd for all such x. Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.

When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. (a) Sketch a possible graph of T as a function of the time t that has elapsed since the faucet was turned on. (b) Describe how the rate of change of T with respect to t varies as t increases. (c) Sketch a graph of the derivative of T.