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]

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = 1 / √(1 − x²)

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]

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