MAM2083F
Vector Calculus for Engineers
Tutorial 1
NOTE: This tutorial revises a few sections of first-year Mathematics which will be used and built on in
second-year. If you have undue difficulty with any of these topics, please revise them in the textbook and
your first-year lecture notes. It’s important to get up to speed before it becomes a problem!
1. (a) Find a vector equation of the line ℓ which passes through the points P(1, −4, 1) and
Q(2, 1, −2).
(b) Describe the line segment PQ (just the part of the line between P & Q) by adding a
condition to your answer in (a).
(c) Does the point (5, 17, −11) lie on ℓ?
(d) What is the angle between OP and OQ, where O is the origin?
(e) Find a non-zero vector perpendicular to both OP and OQ.
(f) Write down a Cartesian equation for the plane φ which passes through P, Q and O.
(g) Find the point of intersection of φ and the line x = (−2, −6, 6) + t (0, −1, 0), t ∈ R.
[Hint: Some answers are very quick if you think — don’t just bash it out!]
2. Let m be the line of intersection of the two planes y = 3 and z = −5.
(a) What is the distance from m to the origin?
(b) Let P be a plane in R3 that contains m. Suppose that this plane P is chosen so that the
distance from P to the origin is a maximum. Find a Cartesian equation for P.
3. Let Φ be the plane in R3 that passes through the points A(3, 3, −5), B(−4, 3, 2) and C(4, −2, 4).
(a) Write down parametric equations for Φ.
(b) Find a vector equation of a line through the origin that is parallel to Φ.
4. Find the maximum and minimum values of f ( x ) = 29 x3 + 12x2 + 2
(a) on the interval (−∞, ∞) ,
(b) on the interval [−72, 0] .
5. Sketch the graphs of the following functions. You are not expected to use derivatives, or to
find maxima or minima, but you must show all the asymptotes.
1
1 + x2
x
(b) y =
1 + x2
(c) y = e1/x
(a) y =
x2
1+x
√
(e) y = x2 − 2x + 2
(d) y =
(f) y = ln(cosh x )
6. Find a Cartesian equation for the curve in R2 defined parametrically by
x = 3 sin t, y = 5 cos t,
0≤t≤π
Sketch this curve. Do these parametric equations define y as a function of x?
1
7. For each constant k, the equation
xy = k
represents a hyperbola that has the x- and y-axes as its asymptotes. You can see this by
noting that if | x | is large, then |y| is small, and if | x | is small, then |y| is large. Now consider
the curve C defined by the equation
( x − y − 4)( x + y − 6) = 1
(a) Can the signs of the terms ( x − y − 4) and ( x + y − 6) change arbitrarily, or are they
linked? Can the terms be zero?
(b) Suppose that both x and y are large and positive (with ( x, y) on C). What can you deduce
about ( x − y − 4) and hence abot ( x + y − 6) ? What does this tell you about such
points?
(c) Suppose this time that x is “very positive" and y is “very negative". (By this we mean that
both | x | and |y| are large, but x > 0 and y < 0). What can you say about ( x − y − 4) ?
What does this tell you about ( x, y) ?
(d) What do you deduce if both x and y are very negative? Or if x is very negative and y is
very positive?
(e) Use this information to sketch the graph of C, showing all its asymptotes.
(f) Let C′ be the curve consisting of the points on C such that x ≥ 5. Suggest parametric
equations for C.
8. Integrate each of the following:
(a)
Z
(b)
Z
(c)
Z
xe x dx
(e)
3
x2 e x dx
(f)
x2 + 1
dx
x+1
Z
x+1
(d)
dx
x2 + 1
(g)
Z π/4
cos x dx
− π/4
Z 3
arctan x
−3 1 + x 4
Z π/2 p
− π/2
dx
1 − cos2 x dx
(Have your answers and reasons checked!)
9. (a) What is the definition of a limit of a function of one variable f ( x )? Look it up.
(b) What is the difference between lim f ( x ) ,
lated?
x → a+
lim f ( x ) , and lim f ( x ) ? Are they re-
x → a−
x→a
(c) What is the definition of continuity for a function of one variable, f ( x )? Look it up.
(d) For each of the following function definitions, determine whether the function exists
and whether it is continuous at the given point, and provide reasons:
2
(i) f ( x ) = 1 + 2x − x2 at x = 1 .
(iii) f ( x ) =
at x = 3 .
(
x
−
3
3
x
if x ≤ 2
(ii) f ( x ) =
at x = 2 .
2
12 − x if x > 2
See optional extra question(s) when the Vula version of the tut is uploaded.
2
MAM2083F
10. Let f ( x ) =
Tut 1
(
ex
a + bx
Optional Extra Questions
2019
if x < 0
if x ≥ 0.
(a) Sketch the graph of y = f ( x ) in the case where a = 0 and b = −1.
(b) For which value(s) of a and b is f a differentiable function? Explain.
11. Find the equation of the line tangent to the curve y = x cos x +
point where x = 0.
12. Calculate the length of the curve y = 1 − 4x2 ,
1
+ tan( x2 ) at the
( x − 1 )3
0≤x≤1.
13. Suppose f ( x ) ≥ 0 for all x ∈ [ a, b]. Give at least three different physical interpretations for
the integral
Z b
a
f ( x ) dx .
14. Recall what you did in question 7(b). Now let the function for curve E be
(y − x )( x + y + 2) = 1 ,
y ≤ −1
(a) Sketch the curve E, showing any asymptotes.
(b) Suggest parametric equations for E.
15. (a) What is the definition of the derivative of a function of one variable f ( x )? Look it up.
(b) What is the definition of differentiability for a function of one variable f ( x )? Look it
up.
(c) Which of the following is/are true? (i) If f ( x ) is continuous at x = a then it is differentiable there. (ii) If f ( x ) is differentiable at x = a then it is continuous there. (iii) If f ( x )
is not continuous at x = a then it is not differentiable there.
(d) For each of the following function definitions, determine whether the function is differentiable at the given point a, providing reasons, and find the derivative f ′ (a) if possible:
(i) f ( x ) = | x + 1| at x = −1 .
(
x2 − 4 if x < 0
(ii) f ( x ) =
at x = 0 .
4 − x2 if x ≥ 0
3
(iii) f ( x ) = x1/3 at x = 0 .
(iv) f ( x ) = 2 − 5x2 at x = a .