3.1.2 Finding Instantaneous Velocity

Question 1

Finding Instantaneous Velocity

Answer 1

• The average rate of change is equal to the slope of the secant line between the two points being considered. The instantaneous rate of change is equal to the slope of the tangent line at the point being considered.
• By examining the average rate of change along an interval of length Δt, you can set the length to be as large or small as you like.
• To find the instantaneous rate, take the limit of the average rate on the interval [t, Δt] as Δtapproaches zero.

Question 2

note

Answer 2

• You have now seen that there is a connection between rates of change and slopes of lines.
• Consider the average rate of change between the point (√6 / 2,20) and a point whose x-coordinate is shifted over by a small offset, Δt. The coordinates of that point are (√6 / 2+Δt, P(√6 / 2 + Δt)). Substitute these values into the formula for the average rate of change.
• To find the instantaneous rate of change, you want to reduce the offset to 0. Set up the limit as Δtapproaches 0.
• Expand the limit expression and simplify wherever possible.
• Cancel the Δtfactors from the numerator and the denominator to remove the indeterminate form.
• Evaluate the expression as Δtapproaches 0.
• Professor Burger was traveling at about 33 mph when he passed the 20 mph speed limit sign

Question 3

The provided graph shows the position function of an automobile. Somewhere on the trip the automobile had a flat and the driver pulled over to the side of the road. What time on the graph is the most likely time that the driver stopped?

Answer 3

Time E

Question 4

The position of an object moving in a straight line is described by the position function p (t). Which of the following expressions is equal to the instantaneous velocity of the object at t = 3?

Answer 4

lim Δt→0 p(3+Δt)−p(3) / Δt

Question 5

Brian has decided to begin a workout program in order to gain weight. His weight is given by the function w (t) where t is the time in days and w (t) is his weight in pounds. Which of the following expressions represents the instantaneous rate of change of his weight when t = 55?

Answer 5

lim Δt→0 w(55+Δt)−w(55) / Δt

Question 6

If g(x)=3x^2+x, simplify the expression 
g(−2+Δx)−g(−2) / Δx.

Answer 6

−11+3Δx, Δx≠0

Question 7

Complete the statement.

Instantaneous rates of change ____.

Answer 7

are found by taking the limit of the average rate of change as Δt goes to 0

Question 8

A bowling ball’s position as it rolls down the lane is described by the position functions(t)=5t−1/8t^2, where t is in seconds and s(t)is in feet. What is the bowling ball’s instantaneous velocity at t=4?

Answer 8

4 ft/sec

Question 9

The provided graph shows the position function of an automobile. Notice that the graph is horizontal between points E and F. What does that mean?

Answer 9

The car was not moving at all.

Question 10

A marathon runner’s distance from the starting point is given by the piecewise function
d = 50(t/8)^2,t≤8
12.5(t−8)+50,t>8
where d is measured in yards and tin seconds.What is the runner’s instantaneous velocity at t=4seconds?

Answer 10

6.25 yards/second

Question 11

A particular bird’s flight position in feet is given by the equation P(t)=12t^2/7+t, where t is the number of seconds that elapse. What is the bird’s instantaneous velocity when t=a?

Answer 11

24a/7+1 ft/sec

Question 12

The provided graph shows the position function of an automobile. Based upon the shape of the graph, at which point was the automobile moving the fastest?

Answer 12

Point B

Question 13

As a car brakes to a stop, its position is given by the function x(t)=20t−2t^2, where t is the time in seconds and x(t) gives the car’s position in feet. What is the car’s instantaneous velocity when t=2?

Answer 13

12 feet/sec

Question 14

If h(t)=e^t+2t^2, simplify the expression h(5+Δt)−h(5)/Δt

Answer 14

e^5+Δt + 20Δt+ 2(Δt)^2 −e^5 / Δt