?At 2:00 PM a car's speedometer reads \(30 \mathrm{mi} / \mathrm{h}\). At 2:10 PM it

The graph of a function is shown. Verify that satisfies the hypotheses of Rolle’s Theorem on the interval . Then estimate the value(s) of that satisfy the conclusion of Rolle’s Theorem on that interval.

Draw the graph of a function defined on such that and the function does not satisfy the conclusion of Rolle’s Theorem on .

The graph of a function is shown. (a) Verify that satisfies the hypotheses of the Mean Value Theorem on the interval . (b) Estimate the value(s) of that satisfy the conclusion of the Mean Value Theorem on the interval . (c) Estimate the value(s) of that satisfy the conclusion of the Mean Value Theorem on the interval .

Draw the graph of a function that is continuous on where and and that does not satisfy the conclusion of the Mean Value Theorem on .

The graph of a function is shown. Does satisfy the hypotheses of the Mean Value Theorem on the interval ? If so, find a value that satisfies the conclusion of the Mean Value Theorem on that interval.

The graph of a function is shown. Does satisfy the hypotheses of the Mean Value Theorem on the interval ? If so, find a value that satisfies the conclusion of the Mean Value Theorem on that interval.

The graph of a function is shown. Does satisfy the hypotheses of the Mean Value Theorem on the interval ? If so, find a value that satisfies the conclusion of the Mean Value Theorem on that interval.

The graph of a function is shown. Does satisfy the hypotheses of the Mean Value Theorem on the interval ? If so, find a value that satisfies the conclusion of the Mean Value Theorem on that interval.

Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolle’s Theorem.

Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolle’s Theorem.

Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolle’s Theorem.

Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolle’s Theorem.

Let . Show that but there is no number in such that . Why does this not contradict Rolle’s Theorem?

Let f(x) = tan x. Show that \(f(0)=f(\pi)\) but there is no number c in \((0, \pi)\) such that \(f^{\prime}(c)=0\). Why does this not contradict Rolle's Theorem?

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. \(f(x)=2 x^{2}-3 x+1, \quad[0,2]\) Equation Transcription: Text Transcription: c f(x)=2x^2-3x+1, 0, 2

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. \(f(x)=x^{3}-3 x+2, \quad[-2,2]\) Equation Transcription: Text Transcription: c f(x)=x^3-3x+2, -2, 2

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f(x)=ln x, [1, 4]

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. \(f(x)=1 / x, \quad[1,3]\) Equation Transcription: Text Transcription: c f(x)=1/x, 1, 3

Find the number \(c\) that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at \((c, f(c))\). Are the secant line and the tangent line parallel? \(f(x)=\sqrt{x}, \quad[0,4]\) Equation Transcription: Text Transcription: c (c, f(c)) f(x)=sqrt x, [0, 4]

Find the number \(c\) that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at \((c, f(c))\). Are the secant line and the tangent line parallel? \(f(x)=e^{-x}, \quad[0,2]\) Equation Transcription: Text Transcription: c (c, f(c)) f(x)=e^-x, [0, 2]

Let \(f(x)=(x-3)^{-2}\). Show that there is no value of c in (1, 4) such that \(f(4)-f(1)=f^{\prime}(c)(4-1)\). Why does this not contradict the Mean Value Theorem?

Let \(f(x)=2-|2 x-1|\). Show that there is no value of \(c\) such that \(f(3)-f(0)=f^{\prime}(c)(3-0)\). Why does this not contradict the Mean Value Theorem? Equation Transcription: Text Transcription: f(x)=2-2x-1 c f(3)-f(0)=f'(c)(3-0)

Show that the equation has exactly one real solution. \(2 x+\cos x=0\) Equation Transcription: Text Transcription: 2x+cos x=0

Show that the equation has exactly one real solution. \(x^{3}+e^{x}=0\)

Show that the equation \(x^{3}-15 x+c=0\) has at most one solution in the interval \([-2,2]\). Equation Transcription: Text Transcription: x^3-15x+c=0 [-2, 2]

Show that the equation \(x^{4}+4 x+c=0\) has at most two real solutions. Equation Transcription: Text Transcription: x^4+4x+c=0

(a) Show that a polynomial of degree 3 has at most three real zeros. (b) Show that a polynomial of degree \(n\) has at most \(n\) real zeros. Equation Transcription: Text Transcription: n n

(a) Suppose that \(f\) is differentiable on \(\mathbb{R}\) and has two zeros. Show that \(f^{\prime}\) has at least one zero. (b) Suppose \(f\) is twice differentiable on \(\mathbb{R}\) and has three zeros. Show that \(f^{\prime \prime}\) has at least one real zero. (c) Can you generalize parts (a) and (b)? Equation Transcription: ? ? Text Transcription: f R f'' f R f''

If \(f(1)=10\) and \(f^{\prime}(x) \geqslant 2\) for \(1 \leqslant x \leqslant 4\), how small can \(f(4)\) possibly be? Equation Transcription: ? ?? Text Transcription: f(1)=10 f'(x)geqslant 2 1 leqslant x leqslant 4 f(4)

Suppose that \(3 \leqslant f^{\prime}(x) \leqslant 5\) for all values of x. Show that \(18 \leqslant f(8)-f(2) \leqslant 30\).

Does there exist a function \(f\)such that \(f(0)=-1\), \(f(2)=4\), and \(f^{\prime}(x) \leqslant 2\) for all \(x\) ? Equation Transcription: , ? Text Transcription: f f(0)= -1, f(2)=4 f'(x) leqslant 2 x

Suppose that f and g are continuous on [a, b] and differentiable on (a, b). Suppose also that f(a) = g(a) and \(f^{\prime}(x)<g^{\prime}(x)\) for a < x < b. Prove that f(b) < g(b). [Hint: Apply the Mean Value Theorem to the function h = f - g.]

Show that sin x < x if \(0<x<2 \pi\).

Suppose is an odd function and is differentiable everywhere. Prove that for every positive number , there exists a number in such that .

Use the Mean Value Theorem to prove the inequality

If ( a constant) for all , use Corollary 7 to show that for some constant .

Let and Show that for all in their domains. Can we conclude from Corollary 7 that is constant?

Use the method of Example 6 to prove the identity.

Use the method of Example 6 to prove the identity. \(2 \sin ^{-1} x=\cos ^{-1}\left(1-2 x^{2}\right), \quad x \geqslant 0\)

At 2:00 PM a car's speedometer reads \(30 \mathrm{mi} / \mathrm{h}\). At 2:10 PM it reads \(50 \mathrm{mi} / \mathrm{h}\). Show that at some time between 2:00 and 2: 10 the acceleration is exactly \(120 \mathrm{mi} / \mathrm{h}^{2}\). Equation Transcription: Text Transcription: 30 mi/h 50/mi h 120 mi/h^2

Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider \(f(t)=g(t)-h(t)\), where \(g\) and \(h\) are the position functions of the two runners.] Equation Transcription: Text Transcription: f(t)=g(t)-h(t) g h

Fixed Points A number \(a\) is called a fixed point of a function \(f\) if \(f(a)=a\). Prove that if \(f^{\prime}(x) \neq 1\) for all real numbers \(x\), then \(f\) has at most one fixed point. Equation Transcription: Text Transcription: a f f(a)=a f'(x)neq 1 x f