4. Applications of Derivatives

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

Exponential growth rates

a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?

Exponential growth rates

b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.

Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.

lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

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

lim_x→∞ ln ((x +1) / (x-1))

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