Final Exam
Question 1 [10 marks]
Choose the correct answer:
(i) Let
f(x) = (
0 x ≤ 0
sin 1
x
x > 0
Which of the following statements about f(x) is true?
(a) limx→0+ f(x) = 0
(b) limx→0− f(x) = 0
(c) limx→0 f(x)
(d) limx→0+ f(x)
(ii) The tangent line to the function f(x) = √
8
x−2
at point (x, y) = (6, 4) is given by
(a) 3y = 6x + 2
(b) y = −
1
2
x + 7
(c) 7y = x + 3
(d) y = 2x + 3
(iii) The function f(x) = x is defined on the interval (0, 1). Which of the following statement is true for
f(x)?
(a) f(x) is not a continuous function
(b) f(x) has a maximum at 1
(c) f(x) has a minimum at 0
(d) f(x) does not have a maximum or minimum value
(iv) The absolute maximum and minimum values of f(x) = 2
3
x − 5 defined on the interval −2 ≤ x ≤ 3 are,
respectively:
(a) (3, −3) and (−2, −19/3)
(b) (−2, −19/3) and (3, −3)
(c) (∞, ∞) and (−∞, −∞)
(d) do not exist
(v) The integral R
tan θ sec2
θdθ evaluates to
(a) 1
2
tan2
θ + C
(b) sec2
θ + 3 tan2
θ sec2
θ + C
(c) sec3
θ
3 + C
(d) sec2
θ
2 + C
1
(vi) The integral R
cos θ(tan θ + sec θ)dθ evaluates to
(a) sec2
θ
(b) sec θ tan θ
(c) − sin θ + cos θ
(d) − cos θ + θ + C
(vii) Which of the following functions have a derivative at every point?
(a) a continuous function
(b) a smooth function
(c) an absolute value function
(d) a sawtooth function
(viii) The length of the curve y =
1
3
(x
2 + 2)3/2
from x = 0 to x = 3 is
(a) 53/6
(b) 12
(c) 32/3
(d) 8
27 (10√
10 − 1)
(ix) The polar coordinates corresponding to the Cartesian coordinates (√
3, −1) and (−
√
3, 1) are, respectively,
(a) (√
2, π/4) and 2√
2, −3π/4
(b) (3, π) and (3, π/2)
(c) (2, 11π/6) and (2, 5π/6)
(d) (5, π − arctan(4/3)) and (13, − arctan(12/5))
(x) A point (2, 7π/3) in polar coordinates has the equivalent Cartesian coordinates
(a) (1,
√
3)
(b) (−1,
√
3)
(c) (1, −
√
3)
(d) (3√
3/2, −3/2)
(xi) Referred to origin O, points A and B have position vectors OA = ˆi + 2ˆj − 2
ˆk and OB = 2ˆi − 3ˆj + 6ˆk.
The value of α for which angles ∠AOP and ∠P OB are equal if OP = (1 + α)ˆi + (2 − 5α)ˆj + (−2 + 8α)
ˆk is
(a) 3/10
(b) 1/
√
6
(c) 1/
√
3
(d) √
2
(xii) The position vector of A is
−→OA = ˆi − ˆj + 2ˆk. The acute angle between −→OA and the y axis is given by
(a) cos θ = 1/
√
6
(b) cos θ = 1/
√
2
(c) cos θ =
√
3/2
(d) cos θ = 1/2
2
(xiii) A square matrix of the second order partial derivatives of a scalar-valued function is a
(a) Jacobian matrix
(b) Hessian matrix
(c) orthogonal matrix
(d) diagonal matrix
(xiv) The second order partial derivative ∂
2
f
∂x2 of the function x cos y + yex
is
(a) yex
(b) − sin y + e
x
(c) cos y + yex
(d) −x sin y + e
x
(xv) Given f(x, y, z) = ln(x + 2y + 3z), ∂f
∂z is
(a) 1
x+2y+3z
(b) 2
x+2y+3z
(c) 3
x+2y+3z
(d) none of the above
(xvi) Given w = xy +
e
y
y2+1 ,
∂
2w
∂x∂y is
(a) y
(b) 1
(c) e
y
(d) 2yey
(xvii) The area of the region R bounded by y = x and y = x
2
in the first quadrant is
(a) 1/3
(b) 1/2
(c) 1/6
(d) 3/2
(xviii) The area of the region R enclosed by the parabola y = x
2 and the line y = x + 2 is
(a) 6
(b) 1/3
(c) 33
(d) 9/2
(xix) The double integral R 3
0
R 2
0
(4 − y
2
)dydx evaluates to
(a) 1
(b) 1/6
(c) 16
(d) 3
3
(xx) The double integral R 1
0
R y
2
0
(3y
3
e
xy)dxdy evaluates to
(a) e − 2
(b) π
2
2 + 2
(c) 8 ln 8 − 16 + e
(d) 3
2
ln 2
Question 2 [5 marks]
(a) [1 mark] Show that limh→0
cosh −1
h = 0.
(b) [1 mark] Evaluate limx→0
tan 3x
sin 8x
.
(c) [1 mark] Differentiate y = (1 + tan4 t
12 )
3
.
(d) [2 marks] Find the critical points for the following functions and use the second derivative test to
determine if they are local minima, maxima or points of inflexion. Assume the derivative is defined over the
entire domain.
(i) y = x
3 + x
2 − 8x + 5
(ii) y =
1
√3
1−x2
Question 3 [5 marks]
(a) [2 marks] The temperature (in ◦C) of a cup of juice is observed to be T(t) = 25(1 − e
−0.1t
) where t is
time. Evaluate the rate of change of T(t) and then use integration to find the change in temperature between
times t = 1 and t = 5. Verify your answer by evaluating T(t) at t = 1 and t = 5.
(b) [2 marks] Find the area contained between the curves y = 3x − x
2 and x + x
2
.
(c) [1 mark] Evaluate R
√
1
16−9×2 dx.
Question 4 [5 marks]
(a) [2 marks] Find the areas of the surfaces generated by revolving the curves below about the x axis:
(i) y = x
3/9, 0 ≤ x ≤ 2
(ii) y =
√
x, 3
4 ≤ x ≤
15
4
(b) [3 marks] Solve the following initial value problems:
(i) x
2y
0 + 2xy = ln x, y(1) = 2
(ii) t
du
dt = t
2 + 3u, t > 0, u(2) = 4
(iii) xy0 = y + x
2
sin x, y(π) = 0
Question 5 [5 marks]
(a) [1 mark] Where does the line −→r = (1, 1, 0) + t(2, 3, 4) meet the plane 2x + y − z = 0?
(b) 1 marks Find the point on the plane x – y + z = 2 that is closest to the point P : (1, 1, −1). (ii)
4
Find the distance from the point P to the plane.
(c) 3 marks Find a vector parametric form of the plane in 3-space that passes through the points (1, 4, 2),
(0, 3, 0), and (−1, 1, 3). (ii) Write the plane −→r = (1, 4, 2) + s(−1, −1, −2) + t(−2, −3, 1) in the scalar form
ax + by + cz = d. (iii) Find a parametric equation for the line in 3-space through the point (0,0,0) and that
is perpendicular to the plane in (ii). (iv) Find the point on the plane that is closest to (0,0,0). (v) Find the
distance from the point (0,0,0) to the plane.
Question 6 [5 marks]
Find all the second order partial derivatives of the following functions:
(i) f(x, y) = x + y + xy
(ii) g(x, y) = x
2y + cos y + y sin x
(iii) f(x, y) = x
3y
5 + 2x
4y
(iv) w =
√
u
2 + v
2
(v) z = arctan x+y
1−xy
Question 7 [5 marks]
Evaluate the following double integrals:
(i) R 1
0
R 1−x
0
(x
2 + y
2
)dydx
(ii) R 1
0
R 1−u
0
(v −
√
u)dvdu
(iii) R 2
0
R 2
x
2y
2
sin xydydx
(iv) R 1
0
R 1
y
(x
2
e
xy)dxdy
(v) R 2
0
R 4−x
2
0
xe2y
4−y
dydx
5
