4. Applications of Derivatives

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

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

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

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = { - 2x if x < 0 ; x if x ≥ 0 ; [-1, 1]

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = 7 -x² ; [-1; 2]

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.        .

a. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval.

x=cos x; (0,π/2)

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

1/203

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).