Midterm Review 3 1 One Type of Integral

Midterm Review 3 -1: One Type of Integral of a Function Deblyn Lawrence

Definite Integral Text Book Definition: l Definite Integral- the definite integral of the function f from x = a to x = b gives a way to find the product of (b-a), even if f(x) is not a constant.

Definite Integral…Again, In English l l Definite Integral- finding the area under the curve and above the x-axis. The problem may show you where you should end the counting by saying “count from x = 1 to x = 7”. You can find this by counting squares (make sure you know the scale of the boxes)

Sample Problems 1. Estimate the definite integral by counting squares on a graph for the following function. f(x) = -0. 1 x^2+7 a. x = 0 to x = 5 b. x = -1 to x = 6

Sample Problems 2. You have been hired by an automobile manufacturer to analyze the predicted motion of a new sports car they are building. When accelerated hard from a standing start, the velocity of the car, v(t) ft/sec, is expected to vary exponentially with time, t seconds, according to the equation. v(t) = 100(1 -0. 9^t) a. Draw the graph of function v in the domain [0, 10]. b. What is the range of the velocity function? c. Approximately how many seconds will it take the car to reach 60 ft/second? d. Approximately how far will the car have traveled when it reaches 60 ft/sec? e. At approximately what rate is the velocity changing when t = 5? f. What special mane is given to the rate of change of velocity?

Problem 1 A: To make graph, first graph on grapher and find coordinate points by using the table function. The definite integral is 30. 775, or 30. 8

Problem 1 B: The definite integral is 41. 7

Problem 2 A: Again, to make graph, first graph on grapher and find coordinate points by using the table function.

Problem 2 B Range: 0 ≤ y ≤ 100. This is the range because the limit of the car’s velocity is 100, as indicated by the original equation.

Problem 2 C: It will take approximately 8. 7 seconds for the car to reach a velocity of 60 ft/sec.

Problem 2 D: The car travels approximately 295 ft when it reaches 60 ft/sec. ** I divided 118 by. 4 because of my scale (4 boxes = 10 ft)

Problem 2 E: Rate of velocity change at t = 5 is 6. 24 (ft/sec)/sec. 1. We only want to pay attention to the ycoordinate. 2. 40. 95 – 34. 39 = 6. 56 (distance from 4 sec to 5 sec) 3. 46. 86 – 40. 95 = 5. 91 (distance from 5 sec to six sec) 4. (6. 56 + 5. 91) = 6. 24 (take the average of the distances)

Problem 2 E, Con’t: Notice the red points.

Problem 2 F Acceleration.