MATH 161, Gateway exam, Sample problems
Experience has shown that some students taking MATH 161 have difficulty with material from prerequisite courses, such as College Algebra, Trigonometry and Calculus I. The course assumes that
this material is known and understood, students with weaknesses in their knowledge of this material
tend to struggle in MATH 161. To enable you to judge your knowledge of this material, MATH 161
will start with a gateway exam, which tests required material from lower level courses. (This gateway
exam will count towards your grade.)
The following problems are chosen to be typical of the kind of problem that will be asked in
the gateway exam, and are given as an aid to your pre-semester review. On day 1 of the coming
semester you must be comfortable with doing these kinds of problems without help of a calculator
(as calculators are not permitted in MATH 161 exams). If even after reviewing the material you have
difficulties with this kind of material, you might benefit from enrolling in MATH 160, Calculus I,
instead.
1) [Long Division of Polynomials]
Divide x 5 +9x 3 −3x 2 +4x −1 by x 2 +2x −1 with remainder. (Determine Quotient and Remainder)
1
x
θ
2) [Linear Equations]
Suppose that
x + 2y + 3z = 25
4x + 5y + 6z = 55
7x + 8y
= 4
Determine the values for x, y and z.
?
a) Determine the (exact) value for the length of
remaining√side as a function in x.
3
, determine sin(θ) and cos(θ) for
b) If x =
2
the indicated angle θ.
6) [Quadratic Equations, Completing the
3) [Algebra]
square]
The notation a! (a-factorial) means the product Complete the square to write x 2 + 10x − 30 in
(2n + 2)!
the form (x + a)2 + b. Determine the roots of
.
1 ⋅ 2 ⋅ ⋯ ⋅ (a − 1) ⋅ a. Simplify
(2n)!
x 2 + 10x − 30.
4) [Inequalities, Sets of Real Numbers]
7) [Continuity, Differentiability]
Determine the set of all real numbers x for which Is the function f (x) = ∣x∣ continuous at x = 0? Is
−3x − 2
∣
∣ < 4.
it differentiable at x = 0?
3
5) [Pythagoras’ theorem, Trigonometric functions]
Consider a right angled triangle with one side of
length 1 and one of length x (where 0 < x < 1):
8) [Definition of the derivative]
Identify the graphs A (green), B (blue) and C
(red) as the graphs of a function and its first and
second derivative.
Sample Gateway Exam MATH 161. Colorado State University, A. Hulpke
12) [Curve Sketching]
Consider the function f (x) = 3x 4 −4x 3 −12x 2 +4.
Determine local and absolute maxima and minima and inflection points of the function. Sketch
the graph of f .
C
13) [Substitution]
Evaluate the integral
B
A
9) [Chain√
Rule, Power Rule]
Let f (x) = sin(1/x). Determine f ′ (x).
14)
[Substitution]
Evaluate the integral
∫ x (x
3
4
− 10)48 dx.
sin(x)
∫ cos (x) dx.
2
15) [Substitution]
√
What is the result, after substituting u = x in
√
sin( x)dx (Only substitute, you don’t need
to integrate fully)?
∫
16) [Definite Integrals]
10) [Chain Rule, Derivatives of Trigonometric Determine the value of the definite integral
5
Functions]
(x + 3)3 dx.
cos(x)
′
0
. Determine f (x).
Let f (x) =
tan(x)
∫
Theorem of Calculus]
11) [Power Rule, Patterns in Higher Deriva- 17) [Fundamental
x2
tives]
Let f (x) =
sin(cos(t))dt. Determine
1
1
Determine the 100th derivative of .
f ′ (x).
x
∫
Sample Gateway Exam MATH 161. Colorado State University, A. Hulpke