4. Applications of Derivatives

Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

Root Finding

5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.

Roots (Zeros)

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

g(t) = 1/(1 − t) + √(1 + t) − 3.1, (−1, 1)

Finding Functions from Derivatives

Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = cos(x²)

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.

c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization

8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)

Dependence on Initial Point

8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations

c. x_0=2

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

a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6

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

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

c. ln x = o(x+1)

90. Find f'(0) for 

f(x) = e^(-1/x²), x≠0

   = 0, x = 0.

91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]

b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.