MAT136F Assignment 1 due Friday, May 14 at 11:59 pm Instructions: • This assignment consists of 7 questions for a total of 37 points. • Your solutions must be submitted on Crowdmark by Friday, May 14...

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MAT136F Assignment 1 due Friday, May 14 at 11:59 pm Instructions: • This assignment consists of 7 questions for a total of 37 points. • Your solutions must be submitted on Crowdmark by Friday, May 14 at 11:59 pm. You should complete this assignment well before 11:59 pm to give yourself time to submit everything. • You may print out this document and write your solutions in the space provided or you may write your solutions on your own blank sheets of paper. • Show all of your work. Unsupported answers will not earn credit. • Have fun! Marking Scheme: Question 1 2 3 4 5 6 7 Points 5 5 6 6 7 5 3 MAT136F Assignment 1 Page 1 of 7 1. Let f(x) be a continuous function. Suppose the following three facts are known: • f(0) = 3 • A left endpoint Riemann sum for f(x) on [0, 8] using 4 subintervals yields a value of 30. • A right endpoint Riemann sum for f(x) on [0, 8] using 4 subintervals yields a value of 48. (a) (2 points) What is the value of 4∑ k=1 f(2k)? Show your work. (b) (3 points) What is f(8)? Show your work. MAT136F Assignment 1 Page 2 of 7 2. Let f(x) = 3x2 + 4x− 1. (a) (3 points) By taking the limit of a Riemann sum, evaluate ∫ 2 0 f(x) dx. Show your work. Hint: you will need to use the following facts: n∑ k=1 1 = n, n∑ k=1 k = 1 2 n(n+ 1), and n∑ k=1 k2 = 1 6 n(n+ 1)(2n+ 1) (b) (2 points) By using the Fundamental Theorem of Calculus, evaluate ∫ 2 0 f(x) dx. Show your work. MAT136F Assignment 1 Page 3 of 7 3. Find the total area of the following regions. Show your work. (a) (2 points) The region in the first quadrant below the function y = 5− |x− 3|. (b) (2 points) The region above f(x) and below the x-axis, where f(x) = { x2 − 1, x ≤ 0 ex − 2, x > 0 . (c) (2 points) The region below y = √ 9− (x− 4)2 and above the x-axis. MAT136F Assignment 1 Page 4 of 7 4. For each of the functions below, find the signed area of the region bounded by the function on the given interval. Show your work. (a) (2 points) f(x) = x sin(x2)√ cos(x2) on [0, √ π 4 ] (b) (2 points) g(x) = sin(x) sin(2x) on [−π, π] (c) (2 points) h(x) = x sin(x) sin(2x) on [−π 4 , π 4 ] MAT136F Assignment 1 Page 5 of 7 5. Let f(t) be a continuous function and define F (x) = ∫ 1+x3 1−x f(t) dt. (a) (2 points) If possible, find a number c ∈ R such that F (c) = 0. If not possible, explain why not. (b) (2 points) Alice claims the average value of f(t) on [0, 2] is 1 2 F (1). Is she correct? Explain your answer. (c) (3 points) For this part only, suppose f(t) = ln(t2 + 1). What is F ′(x)? Show your work. MAT136F Assignment 1 Page 6 of 7 6. Suppose that at time t = 0 water starts flowing out of a large tank. The water pours out at a rate of r(t) = 100t+ 50 t2 + t+ 1 litres/min with t measured in minutes. (a) (1 point) What is the physical interpretation of the statement r(1) = 50? (b) (2 points) How much water flows out of the tank in the first 5 minutes? Show your work. (c) (2 points) How long does it take for 300 litres of water to flow out of the tank? Round your answer to the nearest minute. Show your work. MAT136F Assignment 1 Page 7 of 7 7. (3 points) Let a and b be positive real numbers. Show that the following inequality holds: 0 ≤ ∫ b a | sin t| t dt ≤ ln b− ln a