Calculus Assignment: Functions, Derivatives, and Integration Methods

This calculus assignment presents a series of problems designed to test understanding of fundamental calculus concepts. The assignment begins with questions on sinusoidal functions, including determining the period and analyzing wave characteristics. It then delves into related rates and average rate of change problems, using derivatives to solve for these values. Further, the assignment explores work problems involving spring compression and the application of integration in modeling exponential growth and decay. The document also covers the analysis of wave combinations using graphical and analytical methods. Solutions are provided for each question, demonstrating the application of calculus principles to various mathematical scenarios.

Q1. What is the period of sinusoidal function y=4sin (-2x+7)-1
Answer: The period is determined using the formula 2 π
¿ b∨¿¿ , replacing b with -2 in the period
formula
Period= 2 π
¿−2∨¿ ¿ , solving the equation
The absolute value is determined by the distance between zero and a number thus the distance
between -2 and 0 is 2
The common factor will cancel out 2 π
2 . Diving π by 1, we get π
The period of the sinusoidal wave is thus π
Q2. Consider an aeroplane is pointing towards South and flying with an airspeed of 120m/s.
Simultaneously, there is a steady wind blowing due West with a constant speed of 40 m/s.
a. Make a sketch that shows how to find the resultant velocity of the plane; and
cos θ= 40
126.49 ; θ=71.56 °

tanθ= 120
40 , θ=251.56° which is the direction angle
b. What is the resultant speed of the aeroplane?
126.49 m/s
Q3.1 If f(x) =x3-9x, what is the average rate of change of f over the interval (1, 6)
Answer: The average rate of change of a function is established by evaluating the change
in the values of y of the two pointes and then dividing the resultant by the change in the values of
x of the two points
f ( 0 ) −f ( 6)
( 0 )−(1)
Substituting the equation y=x3-9x for f (0) and f (6) and substituting the x in the function with the
corresponding values of x
{1 ¿ ¿=-8/-1=8
Q3.2 If f(x) =ln(x)/x, find the average rate of change of f(x) between 1 and 1.5
Answer: The average rate of change is determined by
f ( 1.5 ) −f (1)
1.5−1 = ln ( 1.5 ) −ln (1)
0.5 = ln( 0.5)
0.5 =−1.38629
Q3.3. What is the rate of change of the interval π ≤ x ≤ 4 π /3 for the function y=4sin x-7
Answer: f ( 4 π /3 )−f ( π)
( 0 )−(1) = 0.969−2.624
1.047 =−1.581

Q4. A force of 1200 N compresses a spring from its natural length to 18 cm to a length of 16 cm.
how much work is done in compressing it from 16 cm to 14 cm?
Answer: F=k×x
1200=k×0.02
K=1200/0.02=60000 N/m
Work done in compressing it from 16 to 14 cm
X=16-14=2 cm=0.02 m
Work done=1/2×6000×0.022=12 joules
Q5.1 Consider derivative of f is defined as f1(x) =x2/(x-2)3. Find the intervals where f is
increasing
Answer: The intervals are determines using critical points which refer to the points at
which f1 is either 0 or undefined
f1=x2/(x-2)3=x2+2x-3
The critical points are x=-3 and x=1 which can divide the number line in three main groups as
shown
The intervals when f1 is decreasing is thus x<-3

Q5.2 Consider derivative of g is defined as f1(x) =x2/(x+3). Find the intervals where g is
decreasing
Answer: Since g(x) is not continuous at x=1, it is not possible to differentiate it at that point thus
x=1 is one of the critical points. The provided derivative offers the other critical point in which
the negative exponents is preferred instead of the fractions
The critical points are x=-3/2 and x=1
The intervals when g1 is decreasing is thus x<1
Q6. Separate the wave y=5sin (Ɵ+53.13⁰) into distinct component waves using compound angle
identities
sin(θ +53.13° )= sin Ɵ cos 53.13⁰ + cos Ɵ sin 53.13⁰, [Applying the formula of sin (A+B) =sin A
cos B +cos A sin B
¿ 1 ×cos 53.13 ° +0 ×sin 53.13 ° [Since sin Ɵ=1 and cos Ɵ=0]
=cos 53.13⁰+0
=-0.9619
Q7. Formulate predictions of exponential growth and decay models using integration methods by
using an example
Answer: Suppose a model of the growth or decline of a population with differential equation as
shown below is made i.e. the rate of growth is proportional to the quantity present, y can be
solved in the equation

dy
dt =k y
Hence,
1dy
dt y = d ln ( y )
dt = d k t
dt =¿ ln y=k t +C
Thus,
y=eln y=e(k t +C)=e(k t) eC
and the setting A=eC , we find
y ( t ) =A ekt
Take note than y (0) =A
A conclusion can thus be made than y (0)ekt is the solution to the differential equation y (0)
Q8. Model combination of sine waves given below graphically and analyse the variation in the
results between graphical and analytical methods
Y1=sinƟ
Y2=sin (Ɵ-90⁰)
Answer:

Y1=sinƟ is shown in yellow, Y2=sin (Ɵ-90⁰) is shown in yellow.
From the graph it can be noticed that subtraction of 90⁰ to the argument has result in a shift of
the graph to the right by 90⁰. Generally, substituting Ɵ by Ɵ-c shifts the graph to the right by c
units. In this case the graph has been shifted to the right 90⁰