MAST10006 Calculus 2 Semester 1 Assessment, 2019
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Semester 1 Assessment, 2019
School of Mathematics and Statistics
MAST10006 Calculus 2
Question 1 (12 marks)
In this question you must state if you use any standard limits, continuity, l’Hˆopital’s rule or the sandwich theorem. You do not need to justify using limit laws.
(a) Evaluate the following limit
α(l) ╱cot(z) - 、 ′
if it exists. If the limit does not exist, explain why it does not exist.
(b) For a constant c e R, consider the real-valued function f defined by
f (z) =
(i) For which value(s) of c e R is f continuous at z = 0?
(ii) For which value(s) of c e R is f differentiable at z = 0?
Justify your answers.
Question 2 (17 marks)
In this question you must state if you use any standard limits, continuity, l’Hˆopital’s rule, the sandwich theorem or convergence tests for series. You do not need to justify using limit laws.
(a) Consider the sequence with nth term
dn = logVn2 - ,n4 - 2n2、、
Determine lim dn, if it exists. If the limit does not exist, explain why it does not exist.
&
(b) Consider a series dn .
n=1
&
(i) For a positive integer k, define s亿 , the kth partial sum of the series dn . n=1
&
(ii) Using the sequence of partial sums, (s亿 }, define what it means for the series dn
n=1
to converge to the sum s e R.
&
(iii) Let (yn} be a sequence such that n(l) yn = 2 and consider the series (yn - yn+1)
n=1
with nth term dn = yn - yn+1. Determine whether the series converges or diverges. If the series converges, determine its sum.
(c) Determine whether the following series converge or diverge. Justify your answers.
(i) n ╱ 、n
& 6n
n!
n=1
& 2n2 + 4
n3 + 3n
n=1
Question 3 (10 marks)
(a) For z > 1, show that
sinh(arccosh z) =,z2 - 1、
Justify the use of the positive square root if necessary.
(b) Using appropriate substitutions, evaluate the indefinite integral
dz′ z } 1、
Question 4 (10 marks)
(a) Use the complex exponential to evaluate the indefinite integral
e5芒 cos(3t) dt、
(b) Check your answer to Part (a) by using integration by parts to evaluate the same indefinite
integral
e5芒 cos(3t) dt、
Question 5 (10 marks)
(a) Define what it means for a first order ordinary differential equation to be separable (for
example by writing down the standard form of such a differential equation). Then describe the steps needed to solve such a differential equation.
(b) Solve the initial value problem
dy -4 sin(z) cos(z)
=
dz y
with - k z k and initial value y(0) = 2.
Question 6 (11 marks)
During a mining accident waste water polluted with cyanide flows into a storage tank. The waste water contains 1 gram of cyanide per litre, the storage tank has a capacity of 20,000 litres and the time t from the start of the accident is measured in minutes.
At time t = 0 the tank contains 10 ′ 000 litres of unpolluted water with no cyanide in it. During the accident 200 litres of waste water flows into the tank per minute and we assume the waste water mixes uniformly and instantaneously with the other water in the tank. In addition, every minute 100 litres of water flows out of the tank through a drain.
(a) Show that 《 (t), the volume of water in the tank at time t measured in litres, is given by 《 (t) = 10′ 000 + 100t′
for 0 < t < 100 minutes.
(b) Show that z(t), the amount of cyanide in the tank at time t measured in grams, satisfies
the differential equation
dz z
dt 100 + t ′
for 0 < t < 100 minutes.
(c) Solve the initial value problem from Part (b) to determine z(t) for 0 < t < 100 minutes.
(d) The storage tank will overflow after 100 minutes. What is the concentration of cyanide in the water, measured in grams per litre, when the water overflows from the tank?
Question 7 (11 marks)
The population p = p(t) of fish in a fish farm is modelled by the logistic equation
= p V1 - 、- h、
The fish population p(t) is measured in hundreds of fish, the time t is measured in months and h is the harvesting rate, which is constant and measured in hundreds of fish per month (so, for example, p(1) = 2 means there are 200 fish after 1 month and h = 1 means that the fish are harvested at a rate of 100 fish per month).
(a) For a harvesting rate h satisfying 0 k h k 4, determine the equilibrium solutions for the
above model in terms of h.
dp
dt
(c) Classify the equilibrium solutions for Part (a) as either stable or unstable. Justify your answer.
(d) Let p& = lim p(t) be the value of the fish population in the long run. If the fish farmer 芒→&
wants to maintain a fish population of p& = 1 ′ 200 fish in the long run, at what rate should she harvest her fish? Justify your answer.
(e) Suppose that h = 3, so that the fish farmer harvests at a rate of 300 fish per month. For
which values of p(0), the initial fish population, will the long run fish population p& be 1,200 fish? Justify your answer.
Question 8 (12 marks) Consider the second order linear differential equation
d2y dy
dt2 dt
for some function A(t).
(a) Find the solutions to the associated homogeneous differential equation.
(b) Find particular solutions to the inhomogeneous differential equation for the following
functions A(t):
(i) A(t) = 18t2 - 13, (ii) A(t) = 5e2芒 .
(c) Find the general solution to the inhomogeneous differential equation when the function A(t) is given by A(t) = e2芒 - 36t2 + 26.
Question 9 (14 marks) Consider a damped vibrating system, where the displacement z(t) is a function of the time t and where z satisfies the homogeneous differential equation
d2z dz
dt2 dt
(a) Find the general solution to the homogeneous differential equation. (b) Hence for 0 < t k ′3(8π), determine the number of values of t for which z(t) = 0.
(c) A forcing function c sin(2t) with positive parameter c } 0 is introduced to act on the vibrating system from Part (a). The new system is governed by the inhomogeneous dif-
ferential equation
d2z dz
dt2 dt
Find the value of d e R such that zР (t) = d cos(2t) is a particular solution to the inho- mogeneous equation. Hence find the general solution to the inhomogeneous equation.
(d) Identify the steady state component and the transient component of the general solution to the inhomogeneous equation.
(e) What is the value of the parameter c such that 3 is the largest value attained by the
steady state component of the general solution to the inhomogeneous equation? Justify your answer.
Question 10 (9 marks) Let s be a surface in R3 given by z = e,α2 +g2 for (z′ y) e R2; that is s = ← (z′ y′ z) e R3 | z = e,α2 +g2 … 、
(a) Find an expression for the level curve of this surface when z = c. For what value(s) of c
does the level curve exist?
(b) Sketch the level curve for c = e2 . Label each axis intercept with its value.
(c) Sketch the cross section of the surface in the yz plane. Label each axis intercept with its value.
(d) Sketch the surface s in R3. Label each axis intercept with its value.
Question 11 (19 marks) Let f : R2 → R be the real-valued function of two variables defined by the equation
f(z′ y) = 3y2 - 2y3 - 3z2 + 6zy、
(a) Find the gradient of f .
(b) Find the directional derivative of f at (2 ′ 0) in the direction from (2 ′ 0) towards (1 ′ 1).
(c) Find the equation of the tangent plane to the surface z = f(z′ y) at the point where (z′ y) = (2′ 0).
(d) Find the second order partial derivatives fαα , fgg , fαg and fgα of f .
(e) Find all stationary points of f, and classify each point as a local maximum, local minimum
or saddle point.
(f) Evaluate
1 2
f(z′ y) dzdy .
0 0
2022-03-08