MAST10006 Calculus 2 Summer Semester Assessment, 2021
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Summer Semester Assessment, 2021
School of Mathematics and Statistics
MAST10006 Calculus 2
Question 1
Let f (x) be a function with domain [-1, 1]. State the definition of the following:
(a) f is right continunous at x = -1.
f is continunous at x = 0.
f is left continunous at x = 1.
Question 2
In this question you must state if you use standard limits, continuity, l’Hˆopital’s rule, the sandwich theorem or any tests for convergence of series; you do not need to state if you use any limit laws.
Find the following limits, or explain why they do not exist:
(a) lim x
(b)n(l)im-o ╱n tanh ╱sin ╱ 、、、n(1) .
(c)n(l)im-o ╱ tanh(sin n)、.
Question 3
In this question you must state if you use standard limits, continuity, l’Hˆopital’s rule, the sandwich theorem or any tests for convergence of series; you do not need to state if you use any
limit laws.
Determine whether the following series an converge or diverge. Justify your answer.
n=1
o nn
(n + 1)! .
n=1
o
(b) n=1
4n |
n3 + 9n - 8n . |
Question 4
Simplify the following expressions:
(a) arccosh(cosh x) for x < 0.
(b) sech(arcsinh x) for x e R.
Question 5
(a)
Evaluate ╱e2北 cos(2x)、.
(b) Evaluate e2北 cos(2北) d北 via integration by parts.
Question 6
(a)
Make the substitution z = and reduce
dy y3 + 2x2y
=
dx xy2 + x3
to a separable first order ODE on z = z(x). Do not solve it.
Make the substitution z = yk for an approperiate number k and reduce
to a linear first order ODE on z = z(x). Do not solve it.
Question 7
dy 2 cos2 y
(a) Find the general solution.
(b)
(c)
Is there a solution y = y(x) such that y(1) = ? If so, find such a solution; if not,
explain why.
Is there a solution y = y(x) such that y(1) = - ? If so, find such a solution; if not,
explain why.
Question 8
Effluent with pollutant concentration 2 (grams/m3 ) flows into a pond at a constant rate r (m3 /min), where 0 < r s 10. The pond has volume 1000 (m3 ) and initial pollutant 100 (grams). The pollutant mixes quickly and uniformly with pond water and flows out of the pond at rate 10 (m3 /min). Then the amount x (grams) of pollutant in the pond at time t (min) satisfies
dx 10x
dt 1000 - (10 - r)t .
(a) For which value(s) of r does the above ODE have an equilibrium solution? Use the
definition of equilibrium solution to justify your answer.
(b) For r = 8, find the function of the concentration y (grams/m3 ) of pollutant in the pond
with time t (min) before it drains out. Specify the domain of this function.
Question 9
(a)
(b)
Determine the value(s) of α e R such that y = 北α is a solution of the ODE
北2y\\ - 6北y\ - 12y = 0.
Find the general solution of the ODE
北2y\\ - 6北y\ - 12y = 0.
(c)
An object of mass 1 (kg) stretches a spring and is set in motion from certain position. The damping constant of the spring is 2 (N . s/m), and the spring constant in Hooke’s law is 1 (N/m). During the motion there is an external force f (t) (N) acting on the mass such that f (t) = e-t + cos t. Then the displacement y = y(t) of the object from the equilibrium position satisfies
y\\ + 2y\ + y = e-t + cos t.
Find the general solution of y(t) to this equation.
(d)
In part (c), if there is no external force then the spring vibration is in weak damping, or critical damping, or strong damping? Give a reason of your answer in terms of the characteristic equation.
Question 10
Let f : R2 → R, f (x, y) = 3y2 - 2y3 - 3x2 + 6xy .
Find the gradient of f .
Find the directional derivative of f at (1, 1) in the direction from (1, 1) towards (0, 0).
(c)
(d)
Find the equation of the tangent plane to the surface z = f (x, y) at the point where (x, y) = (1, 1).
Find the second order partial derivatives f北北 , fyy , f北y and fy北 of f .
(e) Find all stationary points of f , and classify each point as a local maximum, local mini-
mum or saddle point.
1 1
2
(f) Let F (y) = f(x, y)xe北 dx. Evaluate F (y)dy .
-1 -1
2023-02-15