Solved: A derivative formula a. Use the definition of the

Use definition (1) (p. 127) for the slope of a tangent line to explain how slopes of secant lines approach the slope of the tangent line at a point.

Explain why the slope of a secant line can be interpreted as an average rate of change.

Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.

For a given function f, what does f _ represent?

Given a function f and a point a in its domain, what does f _1a2 represent?

Explain the relationships among the slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point.

Why is the notation dy dx used to represent the derivative?

Give three different notations for the derivative of f with respect to x.

914. Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. f 1x2 = x2 - 5; P13, 42

914. Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. f 1x2 = -3x2 - 5x + 1; P11, -72

914. Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. f 1x2 = -5x + 1; P11, -42

914. Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. f 1x2 = 5; P11, 52

914. Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. f 1x2 = 1 x ; P1-1, -12

914. Equations of tangent lines by definition (1) a. Use definition (1) (p. 127) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. f 1x2 = 4 x2 ; P1-1, 42

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 2x + 1; P10, 12

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 3x2 - 4x; P11, -12

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = -7x; P1-1, 72

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 8 - 2x2; P10, 82

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = x2 - 4; P12, 02

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 1>x; P11, 12

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = x3; P11, 12

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 1 2x + 1 ; P10, 12

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 1 3 - 2x ; Pa -1, 1 5 b

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 1x - 1; P12, 12

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = 2x + 3; P11, 22

1526. Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f 1x2 = x x + 1 ; P1-2, 22

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 8x; a = -3

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = x2; a = 3

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 4x2 + 2x; a = -2

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 2x3; a = 10

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 1 1x ; a = 1 4

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 1 x2; a = 1

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 22x + 1; a = 4

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 23x; a = 12

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 1 x + 5 ; a = 5

2736. Derivatives and tangent lines a. For the following functions and values of a, find f 1a2. b. Determine an equation of the line tangent to the graph of f at the point 1a, f 1a22 for the given value of a. f 1x2 = 1 3x - 1 ; a = 2

3740. Lines tangent to parabolas a. Find the derivative function f  for the following functions f. b. Find an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. c. Graph f and the tangent line. f 1x2 = 3x2 + 2x - 10; a = 1

3740. Lines tangent to parabolas a. Find the derivative function f  for the following functions f. b. Find an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. c. Graph f and the tangent line. f 1x2 = 3x2; a = 0

3740. Lines tangent to parabolas a. Find the derivative function f  for the following functions f. b. Find an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. c. Graph f and the tangent line. f 1x2 = 5x2 - 6x + 1; a = 2

3740. Lines tangent to parabolas a. Find the derivative function f  for the following functions f. b. Find an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. c. Graph f and the tangent line. f 1x2 = 1 - x2; a = -1

A derivative formula a. Use the definition of the derivative to determine d dx 1ax2 + bx + c2, where a, b, and c are constants. b. Let f 1x2 = 4x2 - 3x + 10 and use part (a) to find f _1x2. c. Use part (b) to find f _112.

A derivative formula a. Use the definition of the derivative to determine d dx 11ax + b2, where a and b are constants. b. Let f 1x2 = 15x + 9 and use part (a) to find f _1x2. c. Use part (b) to find f _1-12.

4346. Derivative calculations Evaluate the derivative of the following functions at the given point. 43. y = 1>1t + 12; t = 1 44. y = t - t2; t = 2 45. c = 21s - 1; s = 25 46. A = pr2; r = 3 y = 1>1t + 12; t = 1

4346. Derivative calculations Evaluate the derivative of the following functions at the given point. 43. y = 1>1t + 12; t = 1 44. y = t - t2; t = 2 45. c = 21s - 1; s = 25 46. A = pr2; r = 3 y = t - t2; t = 2

4346. Derivative calculations Evaluate the derivative of the following functions at the given point. 43. y = 1>1t + 12; t = 1 44. y = t - t2; t = 2 45. c = 21s - 1; s = 25 46. A = pr2; r = 3 c = 21s - 1; s = 25

4346. Derivative calculations Evaluate the derivative of the following functions at the given point. 43. y = 1>1t + 12; t = 1 44. y = t - t2; t = 2 45. c = 21s - 1; s = 25 46. A = pr2; r = 3 A = pr2; r = 3

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. For linear functions, the slope of any secant line always equals the slope of any tangent line. b. The slope of the secant line passing through the points P and Q is less than the slope of the tangent line at P. c. Consider the graph of the parabola f 1x2 = x2. For x 7 0 and h 7 0, the secant line through 1x, f 1x22 and 1x + h, f 1x + h22 always has a greater slope than the tangent line at 1x, f 1x22.

Slope of a line Consider the line f 1x2 = mx + b, where m and b are constants. Show that f _1x2 = m for all x. Interpret this result.

4952. Calculating derivatives a. For the following functions, find f  using the definition. b. Determine an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. f 1x2 = 13x + 1; a = 8

4952. Calculating derivatives a. For the following functions, find f  using the definition. b. Determine an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. f 1x2 = 1x + 2; a = 7

4952. Calculating derivatives a. For the following functions, find f  using the definition. b. Determine an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. f 1x2 = 2 3x + 1 ; a = -1

4952. Calculating derivatives a. For the following functions, find f  using the definition. b. Determine an equation of the line tangent to the graph of f at 1a, f 1a22 for the given value of a. f 1x2 = 1 x ; a = -5

5354. Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. a. At which points is the slope of the curve negative? b. At which points is the slope of the curve positive? c. Using AE, list the slopes in decreasing order.

5354. Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. a. At which points is the slope of the curve negative? b. At which points is the slope of the curve positive? c. Using AE, list the slopes in decreasing order.

Power and energy Energy is the capacity to do work, and power is the rate at which energy is used or consumed. Therefore, if E1t2 is the energy function for a system, then P1t2 = E _1t2 is the power function. A unit of energy is the kilowatt-hour (1 kWh is the amount of energy needed to light ten 100-W lightbulbs for an hour); the corresponding units for power are kilowatts. The following figure shows the energy consumed by a small community over a 25-hour period. a. Estimate the power at t = 10 and t = 20 hr. Be sure to include units in your calculation. b. At what times on the interval 30, 254 is the power zero? c. At what times on the interval 30, 254 is the power a maximum?

Population of Las Vegas Let p1t2 represent the population of the Las Vegas metropolitan area t years after 1950, as shown in the table and figure. a. Compute the average rate of growth of Las Vegas from 1970 to 1980. b. Explain why the average rate of growth calculated in part (a) is a good estimate of the instantaneous rate of growth of Las Vegas in 1975. c. Compute the average rate of growth of Las Vegas from 1990 to 2000. Is the average rate of growth an overestimate or underestimate of the instantaneous rate of growth of Las Vegas in 2000? Approximate the growth rate in 2000.

5760. Find the function The following limits represent the slope of a curve y = f 1x2 at the point 1a, f 1a22. Determine a possible function f and number a; then calculate the limit. lim xS2 1 x + 1 - 13 x - 2

5760. Find the function The following limits represent the slope of a curve y = f 1x2 at the point 1a, f 1a22. Determine a possible function f and number a; then calculate the limit. lim hS0 12 + h - 12 h

5760. Find the function The following limits represent the slope of a curve y = f 1x2 at the point 1a, f 1a22. Determine a possible function f and number a; then calculate the limit. lim hS0 12 + h24 - 16 h

5760. Find the function The following limits represent the slope of a curve y = f 1x2 at the point 1a, f 1a22. Determine a possible function f and number a; then calculate the limit. lim xS1 3x2 + 4x - 7 x - 1

Is it differentiable? Is f 1x2 = x2 - 5x + 6 x - 2 differentiable at x = 2? Justify your answer.

Looking ahead: Derivative of xn Use the definition f _1x2 = lim hS0 f 1x + h2 - f 1x2 h to find f _1x2 for the following functions. a. f 1x2 = x2 b. f 1x2 = x3 c. f 1x2 = x4 d. Based on your answers to parts (a)(c), propose a formula for f _1x2 if f 1x2 = xn, where n is a positive integer.

Determining the unknown constant Let f 1x2 = b 2x2 if x 1 ax - 2 if x 7 1. Determine a value of a (if possible) for which f _ is continuous at x = 1.

Let f 1x2 = 1x. a. Find the exact value of f _142. b. Show that f _142 _ f 14 + h2 - f 142 h = 24 + h - 2 h . c. Complete columns 2 and 5 of the following table and describe how 24 + h - 2 h behaves as h approaches 0. d. The accuracy of an approximation is measured by error = _ exact value - approximate value _. Use the exact value of f _142 in part (a) to complete columns 3 and 6 in the table. Describe the behavior of the errors as h approaches 0.

Another way to approximate derivatives is to use the centered difference quotient: f _1a2 _ f 1a + h2 - f 1a - h2 2h . Again, consider f 1x2 = 1x. a. Graph f near the point 14, 22 and let h = 12 in the centered difference quotient. Draw the line whose slope is computed by the centered difference quotient and explain why the centered difference quotient approximates f _142. b. Use the centered difference quotient to approximate f _142 by completing the following table. c. Explain why it is not necessary to use negative values of h in the table in part (b). d. Compare the accuracy of the derivative estimates in part (b) with those found in Exercise 64.

The following table gives the distance f 1t2 fallen by a smoke jumper t seconds after she opens her chute. a. Use the forward difference quotient with h = 0.5 to estimate the velocity of the smoke jumper at t = 2 seconds. b. Repeat part (a) using the centered difference quotient.

The error function (denoted erf 1x2) is an important function in statistics because it is related to the normal distribution. Its graph is shown in the figure, and values at several points are shown in the table. a. Use forward and centered difference quotients to find approximations to d dx 1erf 1x22 ` x = 1 . b. Given that d dx 1erf 1x22 ` x = 1 = 2 e1p , compute the error in the approximations in part (a).