MAT136 Test 1 Version A
A1. [2 marks] A cyclist travels along a street. Their velocity v(t) measured in km per hour is given in the table below, where t is measured in hours. Use the information in the table to write a Riemann sum that approximates the total distance travelled by the cyclist in the first 6 hours. Your Riemann sum should use 3 rectangles and left endpoints. You are NOT expected to use “summation notation”. Your answer can contain “+” and other arithmetic symbols.
A2. [2 marks] Let q(x) be the rate of change of thickness of a wheel mea-sured in mm per km travelled by the wheel. Here x is the distance travelled, measured in km. Write in the box below the unit of measurement of the quantity calculated by the following integral:
A3. [2 marks] Consider the area of the finite region bounded by the graphs of f(x) = x
2 + 3x − 4, g(x) = −x − 4, x = −4 and the y-axis. Write in the box below an integral that you can use to find this area. DO NOT EVALUATE THE INTEGRAL.
A4. [2 marks] Use the graph of y = f(x), provided below, to compute the average value of f(x) on the interval [2, 6]. Write your answer in the box.
A5. [2 marks] Let Find a critical point of G(x).
Write the x-coordinate of this point in the box below. If there is no critical point, write “no critical point”.
A6. [2 marks] Find an anti-derivative of f(x) = x
2 + 6x + 1 whose graph passes through the point (0, 7). Put your answer in the box below.
A7. [2 marks] Evaluate the integral
A8. [2 marks] Evaluate the integral
PART B – Show your work
B1. [4 marks] The graph of the function y = f
′(x) is shown below. On the blank axes, sketch a graph of y = f(x) so that f(0) = 0. Your graph should include any critical points.
B2.(a) [5 marks] Calculate the following integral. Show your work.
B2.(b) [5 marks] Use the method of integration by parts to evaluate the following integral. Show your work.
B3. For all parts of this question, consider the following limit of Riemann sums:
B3.(a) [1 mark] Copy into the box the part of the above expression that is equal to the width of each rectangle in the Riemann sum.
B3.(b) [4 marks] Write an integral that is equal to the expression at the top of this page. DO NOT EVALUATE THE INTEGRAL.
B3.(c) [4 marks] The x-value of the point where we take the height of each rectangle is given by 2 + . Explain in words what is represented by each of 2, 5, i, n, and the expression 2 + .
Note: You will be graded on accuracy, clarity, brevity, and the use of language that would be understandable by another student in these course. These are the same criteria that were discussed in ACT D and the Practice Test.