4. Applications of Derivatives

Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

f'(x) = 2x − 1, P(0,0)

Quadratic approximations

a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:

i. Q(a) = f(a)

ii. Q′(a) = f′(a)

iii. Q″(a) = f″(a).

Determine the coefficients b₀, b₁, and b₂.

2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

a. 10x^4 + 30x + 1

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

lim_x→ 0 (eˣ - 1) / (2x + 5)

112. True, or false? Give reasons for your answers.

c. ln x = o(x+1)

100-m speed The Jamaican sprinter Usain Bolt set a world record of 9.58 s in the 100-meter dash in the summer of 2009. Did his speed ever exceed 30 km/hr during the race? Explain.

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

c. f(x) = 10x^3 + 2x^2, g(x) = e^x

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = x√(1 − x²)

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

lim_x→0 x csc x

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

lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

ln x and log₁₀ x

Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change

Checking the Mean Value Theorem

Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.

f(x) =√(x − 1), [1, 3]

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x/6 - sec x on [0,8]

The Mean Value Theorem                                                                                                                                                                  

 a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

[Technology Exercises] b.Find the solution to as many decimal places as you can.