4. Applications of Derivatives

A graph of ƒ and the lines tangent to ƒ at x = 1, 2 and 3 are given. If x₀ = 3, find the values of x₁, x₂, and x₃, that are obtained by applying Newton’s method. <IMAGE>

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

lim_t→0 (1 - cos 6t) / 2t

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = x⁴ᐟ⁵, [0, 1]

78. Which one is correct, and which one is wrong? Give reasons for your answers.

a. lim (x → 0) (x² - 2x) / (x² - sin x) = lim (x → 0) (2x - 2) / (2x - cos x) = lim (x → 0) 2 / (2 + sin x) = 2 / (2 + 0) = 1

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

lim_x→∞ (e¹/ₓ - 1)/(1/x)

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

99. lim(x→0) (2^sin(x) - 1)/(e^x - 1)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

55. lim (x → ∞) (ln x)^(1/x)

Finding Position from Velocity or Acceleration

Exercises 45–48 give the acceleration a=d²s/dt², initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.

a = 32, v(0) = 20, s(0) = 5

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.

g(x) = x³ - x² - 5x - 3; [-1, 3]

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

60. lim (x → 0) (e^x + x)^(1/x)

"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

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f(x) = x² + 2x, x₀ = 1, dx = 0.1

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

22. The function ln x grows slower than any polynomial Show that ln(x) grows slower as x→∞ than any nonconstant polynomial.

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

x¹⸍² and x¹⸍³