Calculus and Linear Algebra
Math for Computer Science practice questions
and solutions
PROBLEM SETS 1 & 2.
Problem Set 1.
1. Given a quadratic equation of the from ax2 + bx + c = 0, we can solve for x
using the formula
.
Using the above formula, solve the equation 4x2 − 5x − 6 = 0 for x.
2. Find f(x) if f(x + 1) = x2 − 5x + 3.
3. Graph the following functions, and give their domain and range.(a) y = 3x −
2.
(b) y = 4x2 + 3.
4. Graph the following functions, and give their domain and range.(a) y = 2x.
(b) y = 2x+3.
5. Graph the following functions, and give their domain and range.(a) y =
log2(x).
(b) y = log2(x) + 5.
6. Graph the following functions, and give their domain and range.(a) y = sin(x).
(b) y = sin(2x).
7. Graph the following functions, and give their domain and range.(a) y = tan(x).
(b) y = 4tan(x).
8. Simplify the following expressions.
.
9. Make the following computations using right triangles.
(a) For , compute sin(θ), cos(θ), and tan(θ).
(b) For , compute sec(θ), csc(θ), and cot(θ).
, 10. Simplify the following expressions (i.e. write them in terms of elementary
trig functionssin(φ), sin(θ), etc.). (a) sin(θ + φ).
(b) cos(3θ).
1
2 PROBLEM SETS 1 & 2
Problem Set 2. Problems from Lecture 2.
Reading. Quick Calculus, pp. 50–97.
Supplementary reading. Simmons, Chapter 2, sections 2.1–2.5. Read section 2.6 if you are
interested in some applications of the derivative.
1. Compute the following limits.
2. Where are the following functions discontinuous?
3. Use the definition of the derivative to show that for f(x) = ax2 + bx + c, for constants a,b,c
∈ R, the derivative is f0(x) = 2ax + b.
4. Use the definition of the derivative to find the derivative of the function .
5. Use the definition of the derivative and the double angle formula to compute the
derivative of f(θ) = cos(θ).
6. Sketch the graph of the following two functions. For each, state where it is not
differentiable.
(a)
(b) f(x) = |x2 − 9|.
,
7. LetWhat must m and b be for f(x) to be
differ-
entiable at all points?
8. A penny is dropped off a ledge on the World Trade Center in New York City. The ledgeis
1024 feet above the ground. The penny falls a distance of s = 16t2 feet in t seconds. (a)
How long does the penny fall before it hits the ground?
(b) What is the average velocity at which the penny falls during the first three
seconds?9. With the same situation as in Problem [?], answer the following questions.
(a) What is the average velocity at which the penny falls during the last four seconds?
(b) What is the instantaneouse velocity of the penny when it hits the ground?
Math for Computer Science practice questions
and solutions
PROBLEM SETS 1 & 2.
Problem Set 1.
1. Given a quadratic equation of the from ax2 + bx + c = 0, we can solve for x
using the formula
.
Using the above formula, solve the equation 4x2 − 5x − 6 = 0 for x.
2. Find f(x) if f(x + 1) = x2 − 5x + 3.
3. Graph the following functions, and give their domain and range.(a) y = 3x −
2.
(b) y = 4x2 + 3.
4. Graph the following functions, and give their domain and range.(a) y = 2x.
(b) y = 2x+3.
5. Graph the following functions, and give their domain and range.(a) y =
log2(x).
(b) y = log2(x) + 5.
6. Graph the following functions, and give their domain and range.(a) y = sin(x).
(b) y = sin(2x).
7. Graph the following functions, and give their domain and range.(a) y = tan(x).
(b) y = 4tan(x).
8. Simplify the following expressions.
.
9. Make the following computations using right triangles.
(a) For , compute sin(θ), cos(θ), and tan(θ).
(b) For , compute sec(θ), csc(θ), and cot(θ).
, 10. Simplify the following expressions (i.e. write them in terms of elementary
trig functionssin(φ), sin(θ), etc.). (a) sin(θ + φ).
(b) cos(3θ).
1
2 PROBLEM SETS 1 & 2
Problem Set 2. Problems from Lecture 2.
Reading. Quick Calculus, pp. 50–97.
Supplementary reading. Simmons, Chapter 2, sections 2.1–2.5. Read section 2.6 if you are
interested in some applications of the derivative.
1. Compute the following limits.
2. Where are the following functions discontinuous?
3. Use the definition of the derivative to show that for f(x) = ax2 + bx + c, for constants a,b,c
∈ R, the derivative is f0(x) = 2ax + b.
4. Use the definition of the derivative to find the derivative of the function .
5. Use the definition of the derivative and the double angle formula to compute the
derivative of f(θ) = cos(θ).
6. Sketch the graph of the following two functions. For each, state where it is not
differentiable.
(a)
(b) f(x) = |x2 − 9|.
,
7. LetWhat must m and b be for f(x) to be
differ-
entiable at all points?
8. A penny is dropped off a ledge on the World Trade Center in New York City. The ledgeis
1024 feet above the ground. The penny falls a distance of s = 16t2 feet in t seconds. (a)
How long does the penny fall before it hits the ground?
(b) What is the average velocity at which the penny falls during the first three
seconds?9. With the same situation as in Problem [?], answer the following questions.
(a) What is the average velocity at which the penny falls during the last four seconds?
(b) What is the instantaneouse velocity of the penny when it hits the ground?
