4. Applications of Derivatives

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

√146

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

ln 1.05

Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).

Suppose f is differentiable on (-∞,∞) and the equation of the line tangent to the graph of f at x = 2 is y = 5x -3. Use the linear approximation to f at x = 2 to approximate f(2.01).

Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.

Use linear approximation to estimate f (5.1) given that f(5) = 10 and f'(5) = -2.

Suppose f is differentiable on (-∞,∞) and f(5.01) - f(5) = 0.25.Use linear approximation to estimate the value of f'(5).

Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).

Linear approximation Find the linear approximation to the following functions at the given point a.

f(x) = 4x² + x; a = 1

Linear approximation Find the linear approximation to the following functions at the given point a.

g(t) = √(2t + 9); a = -4

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

e⁰·⁰⁶

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

1/³√510

45–46. Linear approximation

a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?

ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).

Estimating speed Use the linear approximation given in Example 1 to answer the following questions.

If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed?