4. Applications of Derivatives

"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)

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)

Finding Position from Velocity or Acceleration

Exercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.

v = 9.8t + 5, s(0) = 10

Finding Position from Velocity or Acceleration

Exercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.

v = sin πt, s(0) = 0

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

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 = 9.8, v(0) = −3, s(0) = 0