4. Applications of Derivatives

Finding Functions from Derivatives

In Exercises 31–36, find all possible functions with the given derivative.

a. y' = 1 / 2√x

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

f(x) = x³ + 4x² + 7, (−∞, 0)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (3 sin 4x) / 5x

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.

f(x) = x² - 6; x₀ = 3

11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.

ƒ(x) = 1 - | x | ; [-1, 1]

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0 csc 6x sin 7x

5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.

a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .

b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

41. lim (x → 0⁺) (ln x)² / ln(sin 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) = 2x + 1

Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?

Roots (Zeros)

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²

Exponential growth rates

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

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

g. x^3 e^(-x)

Use a linear approximation (differentials) to estimate the value of $e^{-0.01}$. Which of the following is the best estimate?