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Mathematical and Computational Sciences

MAT157Y5Y- Term Test 5

Test Booklet

This test contains three (3) questions. You must write your solution to each question on a separate piece of paper. Each question will be individually submitted to Crowdmark, and the transfer process must have started prior to the given deadline.

• You are allowed to use your books and notes.

• You are expected to do the test on your own, without help from any other person.

• You are not allowed to transmit or receive information from any other individual who is not a course instructor, whether in person, digitally, or by some other means.

• You are not allowed to access the internet, except to download and print your test, to submit your test digitally through Crowdmark, to send an email to one of the course instructors, to access course materials, or to post a private message to the instructors through Piazza.

Unless otherwise indicated, you are required to show your work on each problem on this exam. You are expected to provide a clear explanation of your work and reasoning. The following rules apply:

• Organize your work in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive very little credit.

• Explain your work! A correct solution will have a mixture of words and mathematics. You must convey to us that you understand the material, and this is best achieved by providing (short) expla- nations.

• Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work, will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.

• If a question asks you to explain your reasoning, your answers should be short and to the point. Do not provide long, rambling explanations.

1. (i) Suppose that f : [a,b] → R is integrable and g : [a,b] → R is continuously differentiable such that g′ is nowhere zero on [a,b]. Show that there exists c ∈ (a,b) such that

ˆab f(x)g(x)dx = g(a) ˆac f(x)dx + g(b) ˆcb f(x)dx.

Hint: You showed in tutorial that if f : [a,b] → R is continuous and g : [a,b] → R is integrable and sign-semi-definite (non-negative or non-positive), then there exists a point c ∈ (a,b) such that

ˆab f(x)g(x)dx = f(c) ˆab g(x)dx.

You may use this result without proof.

(ii) Suppose that h : [a,b] → R is twice differentiable and satisfies the following properties:

1. h′′ is a continuous function such that h′′ (x) 0 for all x ∈ [a,b]; and

2. There exists an M > 0 such that h′ (x) ≥ M for all x ∈ [a,b]

Show that

ˆab sin(h(t))dt ≤ .

Hint: Multiply and divide the integrand by h′ (t) and use part (i).

2. Each of the following statements is either True or False. If the statement is True, prove it. If the statement is False, provide a counter example. You should justify why your counterexample works, though a full proof is unnecessary.

(i) If f : (0, ∞) → R is a continuous function such that xl f(x) = 0, then ˆ1∞ f(x)dx converges. (ii) If f : (0, ∞) → R is an integrable function, and ˆ1∞ f(x)dx converges, then x f(x) = 0.