2.1.4 Instantaneous Velocity and the Derivative

Question 1

Instantaneous Velocity and the Derivative

Answer 1

• The instantaneous velocity is the derivative with respect to time of the position function .
• The instantaneous velocity of a position function of the form is given by the equation .

Question 2

note

Answer 2

• The instantaneous velocity is the slope of the tangent to the position function. The idea of instantaneous velocity comes from finding the average velocity over smaller and smaller time intervals. In the limit as the time interval approaches zero, the instantaneous velocity is . On the graph, the instantaneous velocity is the line that is tangent to the position curve.The position function of the iguanodon is a straight line through the origin as shown on the graph. The equation for the iguanodon’s motion is x= ct, where c is a constant. The instantaneous velocity is found by taking the derivative of the position function with respect to time:. The iguanodon’s velocity is constant since his motion is uniform.
• A ball tossed in the air does not have uniform motion as shown by the parabolic position curve. One way to find the instantaneous velocity of the ball at, for example t = 1, is to draw a line tangent to the curve (straight line) and measure its slope. This method is somewhat crude in that it depends on the accuracy of your drawing and your ability to measure the slope precisely.

Question 3

The velocity of an object is 12t^3-6t^2+1. Which of the following is the point of the function that describes x as a function of t

Answer 3

3t^4-2t^3+t+c

Question 4

The instantaneous velocity is equal to the slope of the tangent line of a graph of x versus t ____________.

Answer 4

always

Question 5

Which of the following is not a true statement concerning instantaneous velocity?

Answer 5

The expression for x must be in the form at^x so that v=nat^n-1

Question 6

Which of the following statements related to the two graphs is not correct?

Answer 6

The value for x at 3 seconds is not the same value for x at any other time.

Question 7

If the term k is an arbitrary constant, which of the following equations cannot be used to solve for the instantaneous velocity?

Answer 7

x = t + 17y

Question 8

Which of the following is the time derivative of x = at^4-5t^3 +9

Answer 8

4at^3-15t^2

Question 9

The velocity of an object is v=15t^4+7. Which of the following is the position function that describes x as a function of t

Answer 9

x = 3t^5+7t+c

Question 10

Which of the following is the time derivative of x = at^3+7t^2+2t-8?

Answer 10

None of the above

Question 11

Which of the following shows the correct graph for the velocity of an object whose position is described by x = -2t^2+6t

Answer 11

straight decreasing line