Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

The following table gives the ​distance f(t) fallen by a smoke jumper seconds after she opens her chute​. <IMAGE>

a. Use the forward difference quotient with ℎ = 0.5 to estimate the velocity of the smoke jumper at t=2 seconds.

{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by

f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>

Let f (x) = √x.

a. Find the exact value of f' (4).

Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.

Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.

How are the derivatives of sin^−1 x and cos^−1 x related?

Tangent lines Find an equation of the line tangent to the graph of f at the given point.

f(x) = sin⁻¹(x/4); (2,π/6)

Tangent lines Find an equation of the line tangent to the graph of f at the given point.

f(x) = sec⁻¹(e^x); (ln 2,π/3)

Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

a. (f^-1)'(4)

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

c. (f^-1)'(1)

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

d. f'(1)

If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

Find the slope of the curve y=sin^-1 x at (1/2, π/6) without calculating the derivative of sin^-1 x.

Evaluate the derivative of the following functions.

f(x) = sin^-1 2x

Evaluate the derivative of the following functions.

f(x) = sin^-1 (e^-2x)

Evaluate the derivative of the following functions.

f(y) = tan^-1 (2y² - 4)

Evaluate the derivative of the following functions.

f(u) = csc^-1 (2u + 1)

Evaluate the derivative of the following functions.

f(x) = sec^-1 (ln x)

Evaluate the derivative of the following functions.

f(x) = sin(tan^-1 (ln x))

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. d/dx(tan^−1 x) =sec² x

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The lines tangent to the graph of y=sin x on the interval [−π/2,π/2] have a maximum slope of 1.

62​–​65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x) = (x−1) sin^−1 x on [−1,1]

62​–​65. {Use of Tech} Graphing f and f'

b. Compute and graph f'.

f(x) = (x−1) sin^−1 x on [−1,1]

62​–​65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x) = (x−1) sin^−1 x on [−1,1]

62​–​65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(x²−1)sin^−1 x on [−1,1]

62​–​65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(sec^−1 x)/x on [1,∞)

​​​​​47​–​56.​ Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x) = 1/2x+8; (10,4)

13​–​40.​ Evaluate the derivative of the following functions.

f(x) = 1/tan^−1(x²+4)