4. Applications of Derivatives

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→π / 2- (sin x) ^tan x

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_ x→0 ⁺ | ln x | ˣ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ (1 - (3/x))ˣ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→1 ( x- 1)^sinπx

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = 2x + 1

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = sin² x

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = 1/x³

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = 2 - a cos x, a constant

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = (x+4)/(4-x)

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = 3x³ - 4x

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = sin⁻¹ x

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = tan x

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = ln (1 - x)

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

c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).

Approximating changes

Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).