4. Applications of Derivatives

If $f\(\left\)(x\(\right\))=x^3+1$, use the linearization $L\(\left\)(x\(\right\))$ at $a=5$ to approximate $f\(\left\)(5.1\(\right\))$.

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

f(x) = 4x² + x; a = 1

Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.

a. (1.0002)⁵⁰

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

1/³√510

Find the linearizations of

a. tan x at x = -π/4

Graph the curves and linearizations together.

Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.

If $f\left(x\right)=\sqrt{x}+12$, use the linearization $L\left(x\right)$ at $a=16$ to approximate $f\left(16.01\right)$.

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.

a. f(x) = (1 − x)⁶

What is the value of the local linearization of the function $f\left(x\right)=x^2$ at $x=2$, evaluated at $x=2.1$?

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)

Drag racer acceleration The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least _____ mi/hr/s.        .

Shallow-water velocity equation

a. Confirm that the linear approximation to ƒ(x) = tanh x at a = 0 is L(x) = x.

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).

153. The linearization of 2ˣ

a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

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