Year 1 Calculus Practice Exam
Time: 3 hours Full Marks: 100
Section A: Limits and Derivatives
1. (10 points) Evaluate the following limit:
sin(5x)
lim .
x→0 x
2. (10 points) Differentiate the following functions:
(a) f (x) = (3x2 + 5x − 7)4
(b) g(x) = ln(x2 + 1)
3. (10 points) Find the equation of the tangent line to the curve
x2 + 1
y=
x−2
at the point where x = 3.
Section B: Integrals
4. (10 points) Compute: Z
(4x3 − 2x + 1) dx.
5. (10 points) Evaluate the definite integral:
Z π/2
sin2 (x) dx.
0
6. (10 points) Use integration by parts to compute:
Z
xex dx.
Time: 3 hours Full Marks: 100
Section A: Limits and Derivatives
1. (10 points) Evaluate the following limit:
sin(5x)
lim .
x→0 x
2. (10 points) Differentiate the following functions:
(a) f (x) = (3x2 + 5x − 7)4
(b) g(x) = ln(x2 + 1)
3. (10 points) Find the equation of the tangent line to the curve
x2 + 1
y=
x−2
at the point where x = 3.
Section B: Integrals
4. (10 points) Compute: Z
(4x3 − 2x + 1) dx.
5. (10 points) Evaluate the definite integral:
Z π/2
sin2 (x) dx.
0
6. (10 points) Use integration by parts to compute:
Z
xex dx.
