Deck 4: The Derivative in Graphing and Applications

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

Answer 1

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.

Answer 2

False

Question 3

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

A)
B)
C) 0
D)
E)

Answer 3

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.

Question 5

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

Question 6

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

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)

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

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.

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

Question 11

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

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.

Question 13

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

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

Question 15

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

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

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

Question 18

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

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.

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)

Question 21

Does satisfy the hypothesis of the Mean-Value Theorem over the interval [-3, 3]? If so, find all values of C that satisfy the conclusion of the theorem.

Question 22

Does satisfy the hypothesis of the Mean-Value Theorem over the interval [0, 1]? If so, find all values of C that satisfy the conclusion.

Question 23

A cyclist starts from rest and travels 6.25 miles along a straight road in 15 minutes. Use the Mean-Value Theorem to show that at some instant during the trip his velocity was exactly 25 miles per hour.

Question 24

Use Rolle's Theorem to prove that the equation 6x⁵ - 28x³ + 6 = 0 has at least one solution in the interval (0, 1).

Question 25

Does satisfy the hypothesis of the Mean-Value Theorem over the interval [0, 6]? If so, find all values of C that satisfy the conclusion of the theorem.

Question 26

Verify that f(x) = x² + 8 satisfies the hypothesis of the Mean-Value Theorem on the interval [0, 4] and find all values of C that satisfy the conclusion of the theorem.

Question 27

Use Rolle's Theorem to show that f(x) = 4x³ + 5x - 1 does not have more than one real root.

Question 28

Find the value of c in the interval [0, 2 $\pi$ ] that satisfies the Mean Value Theorem. f(x) = 6 cos x

Question 29

Use Rolle's Theorem to show that f(x) = x³ + ax + b, where a > 0, cannot have more than one real root.

Question 30

Does satisfy the hypothesis of the Mean-Value Theorem over the interval [-5, 5]? If so, find all values of C that satisfy the conclusion.

Question 31

Find the value of c in the interval [0, 1] that satisfies the Mean Value Theorem. f(x) = x⁷

Question 32

Verify that f(x) = x³ -3x²- 3x + 1 satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 2] and find all values of C that satisfy the conclusion of the theorem.

Question 33

Find the value of c in the interval [0, 4] that satisfies the Mean Value Theorem. f(x) = 5x² + 20

Question 34

An automobile starts from rest and travels 4.5 miles along a straight road in 6 minutes. Use the Mean-Value Theorem to show that at some instant during the trip its velocity was exactly 45 miles per hour.

Question 35

Find the value of c in the interval [-1, 1] that satisfies the Mean Value Theorem. f(x) = x² - 4x + 3

Question 36

Verify that satisfies the hypothesis of the Mean-Value Theorem over the interval [6, 8] and find all values of C that satisfy the conclusion of the theorem.

Question 37

Approximate by applying Newton's Method to the equation x² - 5 = 0. Use 2 for your initial value and calculate three iterations.

A) 2.2360680
B) 2.2359480
C) 2.2360560
D) 2.236111
E) 2.25

Question 38

Does satisfy the hypothesis of the Mean-Value Theorem over the interval [0, 16]? If so, find all values of C that satisfy the conclusion of the theorem.

Question 39

Find the value of c in the interval [0, $\pi$ ] that satisfies the Mean Value Theorem. f(x) = sin(6x)

Question 40

Find the value of c in the interval [0, 1] that satisfies the Mean Value Theorem. f(x) = x⁴ + 8

Question 41

Use Newton's Method to find the largest positive solution of x⁴ + x³ - 2x²- 5x -7 = 0. Use 4 for your initial value and calculate eight iterations.

A) 3.0777385
B) 2.4705751
C) 2.0313608
D) 2.1446813
E) 2.0413802

Question 42

Use Newton's Method to find the x-coordinate of the intersection of y = x⁴ + 2x³ and y = 2x² + 6x + 3. Use 3 for your initial value and calculate eight iterations.

A) 2.3333333
B) 1.9347826
C) 1.7649117
D) 1.7331057
E) 1.7320508

Question 43

Use Newton's Method to find the largest positive solution of x⁴ + 4x³ -2x² - 9x - 2 = 0. Use 4 for your initial value and calculate eight iterations.

A) 1.6033441
B) 1.5530570
C) 1.8062561
D) 2.2423497
E) 2.9550827

Question 44

Use Newton's Method to approximate the greatest x-coordinate of the intersection of y = x³ - 2x and y = x⁴ + 6x - 4. Use 3 for your initial value and calculate eight iterations.

A) 0.5169625
B) 0.7963536
C) 0.5080644
D) 1.4400943
E) 2.1685393

Question 45

Approximate by applying Newton's Method to the equation x⁵ - 83 = 0. Use -2 for your initial value and calculate nine iterations.

Question 46

Approximate by applying Newton's Method to the equation x³ - 73 = 0. Use 4 for your initial value and calculate five iterations.

Question 47

Use Newton's Method to find the largest positive solution of x⁵ + 6x³-3x² - 9 = 0. Use 3 for your initial value and calculate three iterations.

A) 2.3278689
B) 1.8122908
C) 1.4559632
D) 1.4558412
E) 1.4559512

Question 48

Use Newton's Method to find the largest positive solution of x⁴ + x³ - 6x - 6 = 0. Use 4 for your initial value and calculate eight iterations.

A) 1.8381538
B) 1.9835930
C) 1.8171206
D) 2.3659571
E) 3.0268456

Question 49

Approximate by applying Newton's Method to the equation x³ -29 = 0. Use 3 for your initial value and calculate five iterations.

Question 50

Approximate by applying Newton's Method to the equation x² -44 = 0. Use 6 for your initial value and calculate five iterations.

Question 51

Use Newton's Method to find the largest positive solution of x⁴ - 3x² - 9 = 0. Use 5 for your initial value and calculate eight iterations.

A) 2.2697369
B) 2.5222928
C) 2.2032027
D) 3.0390141
E) 3.8489362

Question 52

Use Newton's Method to find the largest positive solution of x⁴ - 8x² - 9 = 0. Use 4 for your initial value and calculate eight iterations.

A) 3.380208
B) 3.0000000
C) 3.0799923
D) 3.0045383
E) 3.0000157

Question 53

Approximate by applying Newton's Method to the equation x³ - 11 = 0. Use 2.5 for your initial value and calculate four iterations.

A) 2.5
B) 2.253333
C) 2.2243608
D) 2.2239800
E) 2.22398009

Question 54

The equation, x³ - 2x - 19 = 0 has one real solution for 1 < x < 19. Approximate it by Newton's Method. Use 4.75 for your initial value and calculate eight iterations.

Question 55

Use Newton's Method to find the largest solution of x⁵ - 2x⁴ + 5x³ -x² + 7x + 12 = 0. Use 2 for your initial value and calculate six iterations.

A) 1.2151899
B) 0.0852538
C) -1.7308568
D) -1.0071278
E) -1.2897963

Question 56

Use Newton's Method to find the largest positive solution of x³ + x² - 4x - 5 = 3. Use 4 for your initial value and calculate eight iterations.

A) 3.0769231
B) 2.5867842
C) 2.3921405
D) 2.3307530
E) 2.3417624

Question 57

Use Newton's Method to find the largest positive solution of x⁴ + 4x - 5 = 0. Use 4 for your initial value and calculate eight iterations.

A) 1.0000000
B) 1.2164332
C) 1.6112108
D) 2.1938904
E) 2.9730769

Question 58

Use Newton's Method to approximate the solutions of x⁴- 77 = 0. Use 4 for your initial value and calculate four iterations.

A) -3.0108653, 3.0108653
B) -2.9634212, 2.9634212
C) -2.9622573, 2.9622573
D) -3.300781, 3.300781
E) -4, 4

Question 59

Use Newton's Method to find the greatest x-coordinate of the intersection of y = 4x⁴ -24x² and y = 18x² - 16. Use 9 for your initial value and calculate eight iterations.

A) 3.2978272
B) 3.1786982
C) 3.1888714
D) 4.3309497
E) 3.6519603

Question 60

Approximate by applying Newton's Method to the equation x³ - 88 = 0. Use 4 for your initial value and calculate nine iterations.

Question 61

Answer true or false. For the position function graphed, the acceleration at t = 1 is positive.

Question 62

The equation, x³ + x² - 5x - 6 = 0 has one real solution for 1 < x < 6. Approximate it by Newton's Method. Use 3 for your initial value and calculate eight iterations.

Question 63

Let s(t) = t⁴ - 5t + 6 be a position function. The acceleration function a(t) =

A) 4t³ - 5
B) 12t²
C) 4t³ - 5t
D) 12t² - 5
E) 12t³

Question 64

Answer true or false. If a particle is dropped a distance of 624 m. It has a speed of 110.58 m/s (rounded to the nearest hundredth of a m/s) when it hits the ground.

Question 65

The graph represents a position function. Determine what is happening to the velocity at t = 0.

A) It is positive
B) It is negative
C) It is zero
D) There is insufficient information to tell.
E) It is + $\infty$

Question 66

Let s(t) = t⁹ -t be a position function of a particle. At 1 the particle's acceleration is

A) negative
B) positive
C) zero

Question 67

The graph represents a position function. Determine what is happening to the velocity at t = 1.

A) It is negative
B) It is positive
C) It is zero
D) There is insufficient information to tell.
E) It is + $\infty$

Question 68

s(t) = 4t - 3t², t $\ge$ 0. The velocity function is

A) 4 - 3t
B) 4t - 6t²
C) 4 - 6t
D) 8 - 6t
E) 4t - 3t

Question 69

A projectile is dropped, and reaches the ground at 40 m/s. How long does it take the projectile to reach ground? (Assume the position function is , where a = -10 .)

A) 2 s
B) 8 s
C) 4.47 s
D) 4 s
E) 3.16 s

Question 70

Let s(t) = sin 4t be a position function of a particle. At the particle's velocity is

A) Positive
B) Negative
C) Zero

Question 71

The graph represents a velocity function. The acceleration at t = 2 is

A) positive
B) negative
C) zero
D) There is insufficient information to tell.
E) It is + $\infty$

Question 72

Find s when a = 0.

A) 48
B) -48
C) 6
D) -6
E) 0

Question 73

Answer true or false. This can be the graph of a particle's position if the particle is moving to the right at t = 3.3.

Question 74

Let s(t) = 5t⁶ -4t be a position function. Find v when t = 3.

A) 7,290
B) 1,455
C) 7,286
D) 3,633
E) 4

Question 75

The equation, has one real solution for . Approximate it by Newton's Method. Use 2 for your initial value and calculate eight iterations.

Question 76

s(t) = 5t-4t³, t $\ge$ 0. The acceleration function is

A) 5 - 8t
B) 8t
C) 5 - 8t²
D) 24t
E) 5t - 4t

Question 77

Answer true or false. If the graph on the left is a position function, the graph on the right represents the corresponding velocity function.
y = x²

Question 78

The equation, x³ - x² - 4x - 4 = 0 has one real solution for 1 < x < 4. Approximate it by Newton's Method. Use 3 for your initial value and calculate eight iterations.

Question 79

s(t) = 5t⁵- 11. Find t when a = 0.

A) 100
B) -100
C) 5
D) -5
E) 0

Question 80

s(t) = t⁵ -10, t 0. Find s when a = 0.

A) 20
B) -20
C) 10
D) -10
E) 0