Engineering Mathematics: Solved Problems and Examples

This article provides solved problems and examples on various topics in Engineering Mathematics, including sinusoidal functions, aeroplane velocity, derivatives, and work done. It also includes compound angle identities and critical points.

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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 120 m/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
cos θ= 40
126.49

θ=71.56 °
tan θ= 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]?
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)/, find the 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=4 sin x-7
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 of 18 cm to a length of 16 cm.
How much work was 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 the derivative of f is defined as f1=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 the derivative of g is defined as g1=x2/(x+3). Find the intervals where g is
decreasing.
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 =5 sin (θ+53.13 °) 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

References
Spivak, M., 2018. Calculus on manifolds: a modern approach to classical theorems of advanced
calculus. CRC Press
Chartrand, G., Polimeni, A.D. and Zhang, P., 2017. Mathematical Proofs: A transition to
advanced mathematics. Pearson