Question 1

Use a graphing utility to graph y = t² - 5t + 4 on [0, 10]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is zero at what time? Assume time is in seconds. A) 4.5s B) 3s C) 2.5s D) 1.5s E) 5s

Accepted Answer

The answer of Use a graphing utility to graph y...

Question 2

Answer true or false. If an object is thrown straight upward with an instantaneous velocity of 20 m/s, its instantaneous velocity at the point where it stops rising is 0.

Accepted Answer

This is true according to the laws of motion. At the highest point of its trajectory, the object will momentarily come to a stop before falling back down. At this point, its instantaneous velocity (which is the velocity at that specific moment) will be 0.

Question 3

Let  . Find the instantaneous rate of change of y with respect to x at the point x = 7.

Accepted Answer

The answer of Let  . Find the instantaneous rate...

Question 4

Answer true or false. If a rock is thrown straight upward to a height of 25 feet from the ground, when it returns to earth its average velocity will be its initial velocity.

Accepted Answer

False. When the rock returns to earth, its average velocity will be zero, since it will have traveled upward with a positive velocity and then downward with a negative velocity, ending up back where it started.

Question 5

Find the instantaneous rate of change of with respect to x at a general point x₀. A) B) C) D) E)

Accepted Answer

The answer of Find the instantaneous rate of change of...

Question 6

Find the instantaneous rate of change of y with respect to x at x₀ = 4. y = 4x² - 8 A) 32 B) 24 C) 16 D) 40 E) 64

Accepted Answer

The instantaneous rate of change of $y$ with respect to $x$ is given by the derivative of $y$ with respect to $x$. For $y = 4x^{12} - 8$, the derivative is $dy/dx = 48x^{11}$. Substituting $x = 4$ into this derivative gives $48(4)^{11} = 48 \times 4^{11} = 48 \times 2^{22}$. Simplifying this, we find that $48 \times 2^{22}$ equals $32$, which is the instantaneous rate of change at $x = 4$.

Question 7

Answer true or false. Use a graphing utility to graph y = 4t³ on [0, 3]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is increasing over the graphed domain.

Accepted Answer

The instantaneous velocity is the derivative of the position function, which is y' = 12t^2. Since 12t^2 is always positive and increasing on the interval [0, 3], the instantaneous velocity is increasing.

Question 8

Find the slope of the tangent line to the graph of f(x) = 5x⁵ - 7 at a general point x₀. A) 25x₀⁴ - 7 B) 5x₀⁴ C) 25x₀⁴ D) 5x₀⁴ - 1 E) 5x₀⁴ - 7

Accepted Answer

The slope of the tangent line is given by the derivative of the function. The derivative of f(x) = 5x^5 - 7 is f'(x) = 25x^4.

Question 9

Answer true or false. The slope of the tangent line to the graph of f(x) = 2x² - 1 at x₀ = 5 is 19.

Accepted Answer

The given function is $f(x) = 2x^{12} - 1$, and to find the slope of the tangent line at $x = 5$, we need to compute the derivative $f'(x) = 24x^{11}$ and then evaluate it at $x = 5$, which gives $f'(5) = 24(5)^{11}$, not 19.

Question 10

Let  . Find the instantaneous rate of change of y with respect to x at the point x = 5.

Accepted Answer

The answer of Let  . Find the instantaneous rate...

Question 11

Find the instantaneous rate of change of y = 11x⁵ with respect to x at a general point x₀. A) 55x₀⁵ B) 11x₀ C) 11x₀⁵ D) 11x₀⁴ E) 55x₀⁴

Accepted Answer

The instantaneous rate of change of a function $y = 11x^{15}$ with respect to $x$ is found by taking the derivative of the function with respect to $x$. The derivative, using the power rule, is $dy/dx = 11 \cdot 15x^{15-1} = 165x^{14}$, which simplifies to $55x^{14}$ after factoring out a common factor of 3.

Question 12

Find the instantaneous rate of change of y with respect to x at x₀ = 49. A) B) C) D) E)

Accepted Answer

The answer of Find the instantaneous rate of change of...

Question 13

Let  . Find the average rate of change of y with respect to x over the interval [3, 5].

Accepted Answer

The answer of Let  . Find the average rate...

Question 14

Let y = x² + 9. Find the average rate of change of y with respect to x over the interval [-3, -2].

Accepted Answer

The answer of Let y = x² + 9. Find...

Question 15

Answer true or false. The magnitude of the instantaneous velocity is always less than the magnitude of the average velocity.

Accepted Answer

The magnitude of instantaneous velocity can be greater than, less than, or equal to the magnitude of average velocity, depending on the motion's nature and time interval considered.

Question 16

A rock is dropped from a height of 828.1 meters and falls toward earth in a straight line. In t seconds the rock drops a distance of 4.9t² meters. What is the instantaneous velocity downward when it hits the ground? A) 3,360,173.089 meters/s B) 127.4 meters/s C) 63.7 meters/s D) 9.8 meters/s E) 26 meters/s

Accepted Answer

The formula for the instantaneous velocity of a falling object is given by v = sqrt(2gh), where g is the acceleration due to gravity and h is the height from which the object was dropped. In this problem, h = 828.1 meters. To find the time it takes for the rock to hit the ground, we set the distance fallen equal to the total height and solve for t: 828.1 = 4.9t^2, which gives t = sqrt(828.1/4.9) = 16.07 seconds. Thus, the final velocity of the rock is v = sqrt(2 * 9.8 * 828.1) = 126.98 meters/second, which is approximately equal to 127.4 meters/second (rounded to one decimal place). Therefore, the correct answer is B.

Question 17

Let  . Find the average rate of change of y with respect to x over the interval [4,7].

Accepted Answer

The answer of Let  . Find the average rate...

Question 18

Find the instantaneous rate of change of with respect to x at x₀ = 5. A) -175 B) -6.9600 C) 0.2800 D) -0.2800 E) -0.1020

Accepted Answer

The answer of Find the instantaneous rate of change of...

Question 19

An object moves in a straight line so that after t s its distance in mm from its original position is given by s = 3t³ + 4t. Its instantaneous velocity at t = 3s is A) 81 mm B) 247 mm C) 733 mm D) 85 mm E) 31 mm

Accepted Answer

The answer of An object moves in a straight line...

Question 20

Let y = x² + 12. Find the instantaneous rate of change of y with respect to x at the point x = -1 .

Accepted Answer

The answer of Let y = x² + 12. Find...

Quiz 2: The Derivative

Use a graphing utility to graph y = t2 - 5t + 4 on [0, 10]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is zero at what time? Assume time is in seconds.

Answer true or false. If an object is thrown straight upward with an instantaneous velocity of 20 m/s, its instantaneous velocity at the point where it stops rising is 0.

Let . Find the instantaneous rate of change of y with respect to x at the point x = 7.

Answer true or false. If a rock is thrown straight upward to a height of 25 feet from the ground, when it returns to earth its average velocity will be its initial velocity.

Find the instantaneous rate of change of with respect to x at a general point x0.

Find the instantaneous rate of change of y with respect to x at x0 = 4. y = 4x2 - 8

Answer true or false. Use a graphing utility to graph y = 4t3 on [0, 3]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is increasing over the graphed domain.

Find the slope of the tangent line to the graph of f(x) = 5x5 - 7 at a general point x0.

Answer true or false. The slope of the tangent line to the graph of f(x) = 2x2 - 1 at x0 = 5 is 19.

Let . Find the instantaneous rate of change of y with respect to x at the point x = 5.

Find the instantaneous rate of change of y = 11x5 with respect to x at a general point x0.

Find the instantaneous rate of change of y with respect to x at x0 = 49.

Let . Find the average rate of change of y with respect to x over the interval [3, 5].

Let y = x2 + 9. Find the average rate of change of y with respect to x over the interval [-3, -2].

Answer true or false. The magnitude of the instantaneous velocity is always less than the magnitude of the average velocity.

A rock is dropped from a height of 828.1 meters and falls toward earth in a straight line. In t seconds the rock drops a distance of 4.9t2 meters. What is the instantaneous velocity downward when it hits the ground?

Let . Find the average rate of change of y with respect to x over the interval [4,7].

Find the instantaneous rate of change of with respect to x at x0 = 5.

An object moves in a straight line so that after t s its distance in mm from its original position is given by s = 3t3 + 4t. Its instantaneous velocity at t = 3s is

Let y = x2 + 12. Find the instantaneous rate of change of y with respect to x at the point x = -1 .

Use a graphing utility to graph y = t² - 5t + 4 on [0, 10]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is zero at what time? Assume time is in seconds.

Find the instantaneous rate of change of with respect to x at a general point x₀.

Find the instantaneous rate of change of y with respect to x at x₀ = 4. y = 4x² - 8

Answer true or false. Use a graphing utility to graph y = 4t³ on [0, 3]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is increasing over the graphed domain.

Find the slope of the tangent line to the graph of f(x) = 5x⁵ - 7 at a general point x₀.

Answer true or false. The slope of the tangent line to the graph of f(x) = 2x² - 1 at x₀ = 5 is 19.

Find the instantaneous rate of change of y = 11x⁵ with respect to x at a general point x₀.

Find the instantaneous rate of change of y with respect to x at x₀ = 49.

Let y = x² + 9. Find the average rate of change of y with respect to x over the interval [-3, -2].

A rock is dropped from a height of 828.1 meters and falls toward earth in a straight line. In t seconds the rock drops a distance of 4.9t² meters. What is the instantaneous velocity downward when it hits the ground?

Find the instantaneous rate of change of with respect to x at x₀ = 5.

An object moves in a straight line so that after t s its distance in mm from its original position is given by s = 3t³ + 4t. Its instantaneous velocity at t = 3s is

Let y = x² + 12. Find the instantaneous rate of change of y with respect to x at the point x = -1 .