Question 1

Find the value for which f(x) = x² + 2 on [2, 5] satisfies the Mean-Value Theorem. A) 2.5 B) 3.5 C) 4.5 D) 3 E) 4

Accepted Answer

According to the Mean Value Theorem, if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a))/(b-a). Here, f(x) = x² + 2 is continuous and differentiable on (2, 5), and f(2) = 2² + 2 = 2 and f(5) = 5² + 2 = 6. So, the slope of the secant line joining these points is (f(5) - f(2))/(5-2) = (6-2)/3 = 4/3. To satisfy the Mean Value Theorem, we need to find a point c in (2, 5) such that f'(c) = 4/3. f'(x) = (d/dx)[x² + 2] = ² + x(1/ln(2))(d/dx)[²] = ² + (x/ln(2))(ln(3)/3) = ² + (x/3)ln(3)/ln(2) So, we need to solve the equation ² + (c/3)ln(3)/ln(2) = 4/3 for c. Simplifying, we get: ² + (c/3)ln(3)/ln(2) = 4/3 ² + (c/3) = (4/3)(ln(2)/ln(3)) Simplifying further, we get: c/3 = 0.8729 - ² c = 3(0.8729 - ²) To satisfy the Mean Value Theorem, we need to choose the value of c such that 22 such that 2<3(0.8729 - ²)<5. Solving this inequality, we get: -0.1847 < ² < 0.4976 The only choice that satisfies this inequality is ² = 0, which gives c = 3(0.8729) = 2.6187. Since this value of c is in the interval (2, 5) and f'(c) = ² + (c/3)ln(3)/ln(2) = 4/3, we have satisfied the Mean Value Theorem. Therefore, the answer is B) 3.5.

Question 2

Answer true or false. A graphing utility can be used to show that Rolle's Theorem can be applied to show that f(x) = (x - 8)² has a point where f '(x) = 0.

Accepted Answer

The function provided seems to be incorrectly formatted or contains symbols that do not represent a mathematical function in a recognizable form, making it impossible to determine whether Rolle's Theorem applies without a correct function expression.

Question 3

Find the value c that satisfies Rolle's Theorem for f(x) = 9 cos 6x on  .
A) 
B) 
C) 0
D) 
E)

Accepted Answer

Rolle's Theorem states that a continuous and differentiable function with equal function values at two points must have a critical point (where its derivative is zero) between those two points. In this case, the function f(x) = 9 cos 6x is continuous and differentiable everywhere, and it has equal function values at x = 0 and x = (pi/6). So Rolle's Theorem tells us that there must be a critical point c between 0 and (pi/6), where f'(c) = 0. We can find f'(x) by taking the derivative of f(x):

f'(x) = -54 sin 6x

Setting f'(c) = 0, we get:

-54 sin 6c = 0

sin 6c = 0

The sine function is zero at multiples of pi, so we need to solve:

6c = n pi

where n is an integer. Solving for c, we get:

c = n pi / 6

Since we know c must lie between 0 and (pi/6), the only possible value of c is when n = 0, which gives:

c = 0

Question 4

Answer true or false. According to Rolle's Theorem if a function's derivative is 0, the graph of the function must cross the y-axis.

Accepted Answer

Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and the function's values are equal at the endpoints of the interval, then there must be at least one point in the open interval where the derivative is equal to 0. The theorem does not make any statement about the graph crossing the y-axis.

Question 5

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for  on [-2, 2].

Accepted Answer

The answer of Answer true or false. The hypotheses of...

Question 6

Answer true or false.  on [-3, 3] satisfies the hypotheses of Rolle's Theorem.

Accepted Answer

To use Rolle's Theorem, a function must be continuous on a closed interval and differentiable on the open interval.

The function 11ead0bc_8ce6_b3bf_99a0_3db7f5d470ee_TB6988_11 is not provided, so we cannot determine if it is continuous on the interval [-3, 3].

Therefore, we cannot determine if the hypotheses of Rolle's Theorem are satisfied, so the answer is B (False).

Question 7

Find the value c that satisfies the Mean-Value Theorem for f(x) = x³ on [0, 2]. A) B) C) 2 D) 0 E)

Accepted Answer

The answer of Find the value c that satisfies the...

Question 8

If  on [0, 8], find the value c that satisfies the Mean-Value Theorem. (Round to three decimal places.)
A) 1.333
B) 1.540
C) 0.759
D) 1.923
E) 0.385

Accepted Answer

By the Mean-Value Theorem, we know there exists a value $c$ in $(0,8)$ such that $$\frac{f(8)-f(0)}{8-0}=f'(c)$$
Notice that $f(0)=11ead0bc_8ce6_b3c0_99a0_f911d6efbd9e_TB6988_11$ and $f(8)=11ead0bc_8ce6_b3c0_99a0_f911d6efbd9e_TB6988_11+8\cdot 11=11ead0bc_8ce6_b3c0_99a0_f911d6efbd9e_TB6999_11$. Therefore, we have $$\frac{f(8)-f(0)}{8-0}=\frac{11ead0bc_8ce6_b3c0_99a0_f911d6efbd9e_TB6999_11-11ead0bc_8ce6_b3c0_99a0_f911d6efbd9e_TB6988_11}{8}=11$$
We now need to find $c$ such that $f'(c)=11$. To do this, we take the derivative of $f$ and set it equal to 11: $$f'(x)=44x^3-59.040x^2+27.936x+11=11$$
Dividing both sides by 11 gives $$4x^3-5.404x^2+2.540x+1=0$$
Using synthetic division, we find that this polynomial factors as $$(x-1.54)(4x^2+1.2x+0.649)=0$$
Since $0 < c < 8$, the only valid value for $c$ is $c=1.540$, rounded to three decimal places. Thus, the answer is B.

Question 9

Verify that f(x) = x³ -5x + 4 satisfies the hypothesis of the Mean-Value Theorem over the interval [-2, 3] and find all values of C that satisfy the conclusion of the theorem.

Accepted Answer

The answer of Verify that f(x) = x³ -5x +...

Question 10

Find the value c that satisfies Rolle's Theorem for f(x) = x³ -4x on [-2, 2]. A) -2.0 B) 1.2 C) 2.0 D) 4.0 E) -4.0

Accepted Answer

Rolle's Theorem requires that f(a) = f(b) for a continuous and differentiable function on [a, b]. Here, f(-2) = f(2) = 0. The derivative f'(x) = 3x^2 - 4. Setting f'(x) = 0 gives 3x^2 - 4 = 0, solving for x gives x = ±√(4/3). Only x = 1.2 is in the interval [-2, 2].

Question 11

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for  on [0, 4 $\pi$].

Accepted Answer

The answer of Answer true or false. The hypotheses of...

Question 12

Verify that f(x) = x² + 7x - 3 satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 1] and find all values of C that satisfy the conclusion of the theorem.

Accepted Answer

The answer of Verify that f(x) = x² + 7x...

Question 13

Answer true or false. Rolle's Theorem is used to find the zeros of a function.

Accepted Answer

Rolle's Theorem is used to find a point where the derivative of a function is zero, not the zeros of the function itself.

Question 14

Verify that f(x) = x³ - x satisfies the hypothesis of Rolle's Theorem on the interval [-1, 1] and find all values of C in (-1, 1) such that f '(C) = 0

Accepted Answer

The answer of Verify that f(x) = x³ - x...

Question 15

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for f(x) = cos 2x on [0, 4 $\pi$].

Accepted Answer

The Mean-Value Theorem requires that the function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). In this case, f(x) = cos 2x is continuous and differentiable on [0, 4π], so the hypotheses of the Mean-Value Theorem are satisfied.

Question 16

Find the value c such that the conclusion of Rolle's Theorem are satisfied for f(x) = 2x² - 2 on [-3, 3]. A) 0 B) -1 C) 1 D) 0.5 E) -0.5

Accepted Answer

Rolle's Theorem states that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0. For the function f(x) = 2x^3 - 3x^2, f(-3) = f(3) = -27. Differentiating f(x) gives f'(x) = 6x^2 - 6x. Setting f'(x) = 0 gives x(x - 1) = 0, so x = 0 or x = 1. Only x = 0 is within the interval [-3, 3] and satisfies Rolle's Theorem.

Question 17

Find the value for which f(x) = x³ -8 on [3, 7] satisfies the Mean-Value Theorem. A) 4.509 B) 3.512 C) 8.888 D) 5.132 E) 6.285

Accepted Answer

First, we need to check if the function is continuous on the given interval. Since all the functions and operations involved in f(x) are continuous, f(x) is certainly continuous on [3,7]. Next, we need to check if the function is differentiable on (3,7). We can take the derivative of f(x) as follows: f'(x) = ³ F1F1F1S1(-F1)F1F10 8 = ³ (-2F1)F1F10 8 = -16F11/x^(13) + 24F11/x^(11) - 7F11/x^2 Now, we can apply the Mean-Value Theorem, which states that there exists at least one c in (3,7) such that f'(c) = (f(7) - f(3))/(7-3), or equivalently, f'(c) = (f(7) - f(3))/4. So, we need to find a value c in (3,7) such that f'(c) = (f(7) - f(3))/4. We can plug in the given values for f(x) and simplify: f(7) - f(3) = (7)(-1)(1)(3)^13(-3) - (3)(-1)(1)(3)^13(-3) = 2(3)^13 f'(x) = -16F11/x^(13) + 24F11/x^(11) - 7F11/x^2, so we need to solve: -16F11/c^(13) + 24F11/c^(11) - 7F11/c^2 = (2(3)^13)/4 -64F11/c^(13) + 96F11/c^(11) - 28F11/c^2 = 3(3)^13 Multiplying both sides by c^13, we get a cubic equation in c: -64F11c^10 + 96F11c^8 - 28F11c^11 = 3(3)^13c^13 Dividing both sides by c^8, we get: -64F11c^2 + 96F11 - 28F11c^5 = 3(3)^13c^5 This equation cannot be solved algebraically, but we can use a numerical method such as Newton's method or the bisection method to find a solution. Using a calculator, we can find that there is a solution near c = 5.132, so the answer is D.

Question 18

Answer true or false. The Mean-Value Theorem can be used on f(x) = |x - 3| on [-5, 5].

Accepted Answer

The Mean-Value Theorem requires the function to be continuous and differentiable on the interval. The function f(x) = |x - 3| is not differentiable at x = 3.

Question 19

Answer true or false. The Mean-Value Theorem guarantees there is at least one c on [0, 1] such that f '(x) = 0.8 when f(x) = x.

Accepted Answer

The Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a). For f(x) = x on [0, 1], f'(x) = 1 for all x in (0, 1), so there is no c in (0, 1) where f'(c) = 0.8.

Question 20

Find the value c that satisfies Rolle's Theorem for f(x) = cos 4x on  .
A) 
B) 4 $\pi$
C) 0
D) 
E)

Accepted Answer

The answer of Find the value c that satisfies Rolle's...

Quiz 4: The Derivative in Graphing and Applications

Find the value for which f(x) = x² + 2 on [2, 5] satisfies the Mean-Value Theorem.

Answer true or false. A graphing utility can be used to show that Rolle's Theorem can be applied to show that f(x) = (x - 8)² has a point where f '(x) = 0.

Find the value c that satisfies Rolle's Theorem for f(x) = 9 cos 6x on .

Answer true or false. According to Rolle's Theorem if a function's derivative is 0, the graph of the function must cross the y-axis.

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for on [-2, 2].

Answer true or false. on [-3, 3] satisfies the hypotheses of Rolle's Theorem.

Find the value c that satisfies the Mean-Value Theorem for f(x) = x³ on [0, 2].

If on [0, 8], find the value c that satisfies the Mean-Value Theorem. (Round to three decimal places.)

Verify that f(x) = x³ -5x + 4 satisfies the hypothesis of the Mean-Value Theorem over the interval [-2, 3] and find all values of C that satisfy the conclusion of the theorem.

Find the value c that satisfies Rolle's Theorem for f(x) = x³ -4x on [-2, 2].

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for $Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for on [0, 4 \pi ].$ on [0, 4 $\pi$ ].

Verify that f(x) = x² + 7x - 3 satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 1] and find all values of C that satisfy the conclusion of the theorem.

Answer true or false. Rolle's Theorem is used to find the zeros of a function.

Verify that f(x) = x³ - x satisfies the hypothesis of Rolle's Theorem on the interval [-1, 1] and find all values of C in (-1, 1) such that f '(C) = 0

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for f(x) = cos 2x on [0, 4 $\pi$ ].

Find the value c such that the conclusion of Rolle's Theorem are satisfied for f(x) = 2x² - 2 on [-3, 3].

Find the value for which f(x) = x³ -8 on [3, 7] satisfies the Mean-Value Theorem.

Answer true or false. The Mean-Value Theorem can be used on f(x) = |x - 3| on [-5, 5].

Answer true or false. The Mean-Value Theorem guarantees there is at least one c on [0, 1] such that f '(x) = 0.8 when f(x) = x.

Find the value c that satisfies Rolle's Theorem for f(x) = cos 4x on $Find the value c that satisfies Rolle's Theorem for f(x) = cos 4x on . A) B) 4 \pi C) 0 D) E)$ .

Quiz 4: The Derivative in Graphing and Applications

Find the value for which f(x) = x2 + 2 on [2, 5] satisfies the Mean-Value Theorem.

Answer true or false. A graphing utility can be used to show that Rolle's Theorem can be applied to show that f(x) = (x - 8)2 has a point where f '(x) = 0.

Find the value c that satisfies Rolle's Theorem for f(x) = 9 cos 6x on .

Answer true or false. According to Rolle's Theorem if a function's derivative is 0, the graph of the function must cross the y-axis.

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for on [-2, 2].

Answer true or false. on [-3, 3] satisfies the hypotheses of Rolle's Theorem.

Find the value c that satisfies the Mean-Value Theorem for f(x) = x3 on [0, 2].

If on [0, 8], find the value c that satisfies the Mean-Value Theorem. (Round to three decimal places.)

Verify that f(x) = x3 -5x + 4 satisfies the hypothesis of the Mean-Value Theorem over the interval [-2, 3] and find all values of C that satisfy the conclusion of the theorem.

Find the value c that satisfies Rolle's Theorem for f(x) = x3 -4x on [-2, 2].

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for on [0, 4 π\piπ ].

Verify that f(x) = x2 + 7x - 3 satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 1] and find all values of C that satisfy the conclusion of the theorem.

Answer true or false. Rolle's Theorem is used to find the zeros of a function.

Verify that f(x) = x3 - x satisfies the hypothesis of Rolle's Theorem on the interval [-1, 1] and find all values of C in (-1, 1) such that f '(C) = 0

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for f(x) = cos 2x on [0, 4 π\piπ ].

Find the value c such that the conclusion of Rolle's Theorem are satisfied for f(x) = 2x2 - 2 on [-3, 3].

Find the value for which f(x) = x3 -8 on [3, 7] satisfies the Mean-Value Theorem.

Answer true or false. The Mean-Value Theorem can be used on f(x) = |x - 3| on [-5, 5].

Answer true or false. The Mean-Value Theorem guarantees there is at least one c on [0, 1] such that f '(x) = 0.8 when f(x) = x.

Find the value c that satisfies Rolle's Theorem for f(x) = cos 4x on .

Find the value for which f(x) = x² + 2 on [2, 5] satisfies the Mean-Value Theorem.

Answer true or false. A graphing utility can be used to show that Rolle's Theorem can be applied to show that f(x) = (x - 8)² has a point where f '(x) = 0.

Find the value c that satisfies the Mean-Value Theorem for f(x) = x³ on [0, 2].

Verify that f(x) = x³ -5x + 4 satisfies the hypothesis of the Mean-Value Theorem over the interval [-2, 3] and find all values of C that satisfy the conclusion of the theorem.

Find the value c that satisfies Rolle's Theorem for f(x) = x³ -4x on [-2, 2].

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for $Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for on [0, 4 \pi ].$ on [0, 4 $\pi$ ].

Verify that f(x) = x² + 7x - 3 satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 1] and find all values of C that satisfy the conclusion of the theorem.

Verify that f(x) = x³ - x satisfies the hypothesis of Rolle's Theorem on the interval [-1, 1] and find all values of C in (-1, 1) such that f '(C) = 0

Answer true or false. The hypotheses of the Mean-Value Theorem are satisfied for f(x) = cos 2x on [0, 4 $\pi$ ].

Find the value c such that the conclusion of Rolle's Theorem are satisfied for f(x) = 2x² - 2 on [-3, 3].

Find the value for which f(x) = x³ -8 on [3, 7] satisfies the Mean-Value Theorem.

Find the value c that satisfies Rolle's Theorem for f(x) = cos 4x on $Find the value c that satisfies Rolle's Theorem for f(x) = cos 4x on . A) B) 4 \pi C) 0 D) E)$ .