MATH1020 Calculus for Engineers - Workshop 1
1
Calculus for Engineers
Workshop 1
Learning outcomes for this session
At the end of this session, you should be able to
1. Understand exponential and logarithmic functions.
2. Solve equations involving exponential and logarithmic functions.
3. Solve equations involving trigonometric expressions
4. Solve problems involving trigonometric expressions.
Overview
We will work through and discuss exercises based on your prior knowledge of logarithmic,
exponential functions, and trigonometry. It may be a good idea to read the revision notes
that accompany this workshop before attempting the exercises. A good understanding of
these topics is essential for our later work.
Exercises
All questions referring to a Section are from the textbook by Singh.
1. Exercise 5(c) (p.252): 18, 19.
2. Miscellaneous Exercise 5 (p. 268): 2, 3.
3. Use the properties of logarithms to expand the quantity
q √
q
x−1
4
(b) ln(s t u)
(a) log10 x+1
4. Express the quantity as a single logarithm.
(a) log10 4 + log10 a − 13 log10 (a + 1)
(b)
1
3
ln(x + 2)3 + 12 [ln x − ln(x2 + 3x + 2)2 ]
5. Find the domain of the following functions:
2
1 − ex
(a) f (x) =
1 − e1−x2
(b) g(x) =
1+x
ecos x
(c) h(x) = ln x + ln(2 − x)
6. Solve each equation for x:
(a) e2x − ex − 6 = 0
(b) ln(2x + 1) = 2 − ln x
(c) 10(1 + e−x )−1 = 3
MATH1020 Calculus for Engineers - Workshop 1
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7. Stronium-90 has a half-life of 28 days.
(a) A sample has a mass of 50 mg initially. find a formula for the mass remaining
after t days.
(b) Find the mass remaining after 40 days.
(c) How long does it take the sample to decay to a mass of 2 mg?
8. Sketch the following graphs over the domain [0, 2π]
π
(a) y = 3 sin x +
−3
4
π
(b) y = − cos 2x −
3
9. Find the exact trigonometric ratios for θ =
2π
.
3
10. Prove the identity: (sin x + cos x)2 = 1 + sin 2x
11. Compute all solutions for θ ∈ [0, 2π], giving exact solutions in radians.
(a) 2 cos 3θ = 1
(b) 2 sin θ tan θ − tan θ = 1 − 2 sin θ
(c) sin θ = tan θ
12. Find all values of x in the interval [0, 2π] that satisfy the inequality.
(a) 2 cos x + 1 > 0
(b) sin x > cos x
13. Consider a triangle ABC, where angle A is 35◦ , angle B is 65◦ and side b (the side
opposite angle B ) is 18.5cm. Solve for all unknown sides and angles.
14. To find the distance across a lake, a surveyor starts at one end of the lake and walks
245 metres. From that point, the surveyor turns 110◦ and walks 270 metres to the
other end of the lake. Approximately how long is the lake?
15. Prove the Law of Cosines: If a triangle has sides with lengths a, b, and c and θ is
the angle between the sides with lengths a and b, then
c2 = a2 + b2 − 2ab cos θ
(Hint: introduce a coordinate system so that θ is in standard position. Express x
and y in terms of θ and then use the distance formula to compute c.)
16. The figure below shows a cross-section of a component that is made from a round
bar. If the diameter of the bar is 74mm, calculate the dimension x.
MATH1020 Calculus for Engineers - Workshop 1
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Useful formulae:
LOGARITHMS & EXPONENTIALS
y
loga x = y ⇐⇒ a = x,
loga (xy) = loga x + loga y, loga
loga (xr ) = r loga (x),
loga (ax ) = x,
(ab )c = abc ,
ab ac = ab+c ,
x
y
!
= loga x − loga y
aloga x = x
1
a−b = b
a
STANDARD ANGLES
θ (rad)
sin θ
0
0
cos θ
1
π
6
1
√2
3
2
π
4
√1
2
√1
2
π
√3
3
2
1
2
π
2
1
0
TRIG IDENTITIES
sin2 θ + cos2 θ = 1,
tan2 θ + 1 = sec2 θ,
1 + cot2 θ = csc2 θ
sin2 θ = 12 (1 − cos 2θ),
cos2 θ = 12 (1 + cos 2θ)
2 tan θ
sin 2θ = 2 sin θ cos θ,
cos 2θ = cos2 θ − sin2 θ,
tan 2θ =
1 − tan2 θ
1
sin A cos B = 2 [sin(A + B) + sin(A − B)]
cos A cos B = 12 [cos(A + B) + cos(A − B)]
sin A sin B = 21 [cos(A − B) − cos(A + B)]
sin A + sin B = 2 sin 12 (A + B) cos 12 (A − B)
sin A − sin B = 2 cos 12 (A + B) sin 21 (A − B)
cos A + cos B = 2 cos 12 (A + B) cos 12 (A − B)
cos A − cos B = 2 sin 12 (A + B) sin 12 (B − A)
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan A ± tan B
tan(A ± B) =
1 ∓ tan A tan B
(Answers: 3.(a)
√
1
2
log10 (x − 1) − 12 log10 (x + 1), (b) 4 ln s + 12 ln t + 14 ln u; 4.(a) log10
4a
√
,
3
a+1
x
(b) ln x+1
; 5.(a) Df = {x|x 6= ±1}, (b) Dg = (−∞, ∞), (c) Dh = (0, 2); 6.(a) ln 3, (b)
√
√
−1+ 1+8e2
3
2π
3
−t/28
,
(c)
ln
;
7.(a)
50·2
,
(b)
≈
18.6
mg,
(c)
≈
130
days;
9.
sin
=
, cos 2π
=
4
7
3
2
3
√
1
2π
2π
2
2π
2π
1
π 5π 7π 11π 13π 17π
√
√
− 2 , tan 3 = − 3, csc 3 = 3 , sec 3 = −2, cot 3 = − 3 ; 11.(a) 9 , 9 , 9 , 9 , 9 , 9 ,
(b) 3π
, 7π , π , 5π , (c) 0,π,2π; 12.(a) 0 ≤ x < 2π
, 4π
< x ≤ 2π, (b) π4 < x < 5π
;13.
4 4 6 6
3
3
4
◦
6
C = 80 , a ≈ 11.7 cm,c ≈ 20.1 cm; 14. ≈ 296m; 16. 24mm)