Calculus 1501 - Homework #1
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Calculus 1501 - Homework #1
Each question should be submitted on Gradescope (www.gradescope.ca) with your handwritten work clearly shown, and it is expected that homework is done individually.
Remember to justify your calculations and conclusions. A poorly justi
ed solution will not receive many marks. A solution with just a 
nal answer and no work shown will recieve 0. Be sure to clearly state your reasoning "since the function satis
es the conditions of. . .", " . . .which we evaluate using the identity. . .". Calculators are not allowed in this course, so there may be a loss of marks if a simipli
cation is done without showing your rough work.
This assignment is due by 11:00 pm Eastern Time on Thursday July 14th , and each question is to be submitted via Gradescope.
Question # 1. (6 marks)
(a) Prove that lim 3x + 5 = 2 by using the formal e - 6 de
nition we learned.
x → _ 1
6x
x →2 2x + 5
(c) Suppose have some function, f (x), that is de
ned for all real numbers. We are told lim f (x) = -3 . Explain x →0
how we know that there exists a 6 > 0 so that if |x| < 6 then
-4 < f (x) < -2
Question # 2. (4 marks)
(a) Use the substitution, u = ^35 - 2x, to integrate
(b) Suppose we know that f/ (x) = x . Simplify
x ^35 - 2xdx .
sin(e )dxx using the function f (x) in your expression.
Question # 3. (6 marks) For both of these questions, you must apply at least one result learned in section 1A, 1B, or 1C. There may be other ways to answer these questions, but you will not receive marks if you do not correctly use a result from one of those three sections. Be sure to clearly indicate your application (and why it applies) to avoid confusion.
(a) Show that it is impossible to have a function, f (x), with domain (-o, o), which satis
es all of the following
conditions:
(i) f (x) continuous over [3, 7]
7
(ii) f (x)dx = 2
3
(iii) f (1) = 1
(iv) f (x) strictly increasing over the real numbers
(b) Show that if p(x) is a polynomial and that for real numbers a < b < c we have p(a) = p(b) = p(c) then
p\\ (x) = 0 has at least one solution.
Question # 4. (8 marks)
(a) Complete the following integrations
2
(i) x2 cos(2x)dx (ii) ln(x)dx 1
(b) Using integration by parts on xn ^3x + 1dx, demonstrate the reduction formula,
xn ^3x + 1dx = 
- 
xn _1 ^3x + 1dx
(c) Suppose we are given a function, g(x), which has the property, 尸xg(x) - sec ╱x + g(x)、]\ = xg\ (x) . Using this knowledge, integrate
g(x)dx
expressing your answer in terms of g(x) . Hint: Don't bother trying to 
nd g(x) . It can't be determined with the calculus you know!
Question # 5. (6 marks)
(a) Complete the following integrations
|
(i) |
dx |
(ii) |
_1 x2 - 4x + 3 |
the number of each block, and brie
y explain how that block would be used. You do not actually need to
complete the integration.
Question # 6. (8 marks)
(a) Complete the following integrations
(i) tan3 (θ) sec3 (θ)dθ
π
(ii) sin2 (θ)dθ
0
(iii)
dx
x
(b) Suppose that we have two continuous, di
erentiable functions, f(x) and g(x), and two real numbers a < b,
with the properties that, f\ (x) = g(x), f(a) = -1, f(b) = 2, and ╱g(x)、2 = ╱ f(x)、3 + 1 . Evaluate,
ab f(x) ╱g(x)、3 dx
2022-07-12