4. Applications of Derivatives

Roots (Zeros)

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

i. y = x² − 4

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

Roots (Zeros)

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

f(x) = x⁴ + 3x + 1, [−2, −1]

Roots (Zeros)

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

f(x) = x³ + 4x² + 7, (−∞, 0)

Roots (Zeros)

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

g(t) = √t + √(1 + t) − 4, (0, ∞)

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

Roots (Zeros)

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

r(θ) = 2θ − cos²θ + √2, (−∞, ∞)

Finding Functions from Derivatives

Suppose that f(−1) = 3 and that f'(x) = 0 for all x. Must f(x) = 3 for all x? Give reasons for your answer.

Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

a. f(0) = 0

Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

b. f(1) = 0

Finding Functions from Derivatives

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

a. y′ = x

Finding Functions from Derivatives

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

b. y′ = x²

Finding Functions from Derivatives

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

a. y′ = −1 / x²

Finding Functions from Derivatives

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

a. y' = 1 / 2√x

Finding Functions from Derivatives

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

c. y' = sin (2t) + cos (t/2)