Calculus I
Prerequisite Review - ANSWERS
1. Radicals and Exponents
(i) a. √𝑥𝑥
3
b. √𝑥𝑥
f. √𝑥𝑥 3 or �√𝑥𝑥�
(ii) a. 𝑥𝑥 5/3
3
b.
4
4
55
5 5
c. √𝑥𝑥 3 or � √𝑥𝑥�
g.
1
d.
√𝑥𝑥 3 or 7 � √𝑥𝑥�
7
c. 𝑥𝑥1/4
𝑥𝑥 2/3
3
d.
3
5
1
√𝑥𝑥 4
h.
6
or
1
�𝑥𝑥 5
5
1
� √𝑥𝑥�
or
e.
4
6
1
� √𝑥𝑥�
1
√𝑥𝑥 3
i.
5
5
or
1
√𝑥𝑥 12
1
�√𝑥𝑥�
or
3
5
1
� √𝑥𝑥�
12
1
𝑥𝑥 9/5
2. Factoring with negative and rational exponents
a. 𝑥𝑥 4 (3𝑥𝑥 + 5)
d. 18𝑥𝑥 −3 (3 + 𝑥𝑥 5 ) =
g. 4𝑥𝑥 −1/2 (𝑥𝑥 2 + 3) =
b. −15𝑥𝑥(3𝑥𝑥 − 1)
18�3+𝑥𝑥 5 �
𝑥𝑥 3
4�𝑥𝑥 2 +3�
j. 𝑥𝑥 1⁄3 (𝑥𝑥 + 3)(𝑥𝑥 + 2)
𝑥𝑥 1/2
l. (3𝑥𝑥 1⁄3 − 1)(𝑥𝑥 1⁄3 − 7)
3. Lines: graphing
c. 4𝑥𝑥 −1 (𝑥𝑥 4 + 3) =
e. 4𝑥𝑥 −7 (1 + 11𝑥𝑥 5 ) =
h. 12𝑥𝑥 −5/2 (3 + 𝑥𝑥 3 ) =
k. 𝑥𝑥 −2⁄3 (3𝑥𝑥 − 1)(2𝑥𝑥 − 3) =
m. 3(2𝑒𝑒 𝑥𝑥 − 1)(𝑒𝑒 𝑥𝑥 + 3)
4�𝑥𝑥 4 +3�
4�1+11𝑥𝑥 5 �
𝑥𝑥 7
12�3+𝑥𝑥 3 �
𝑥𝑥 5/2
𝑥𝑥
(3𝑥𝑥−1)(2𝑥𝑥−3)
𝑥𝑥 2/3
f. 4𝑥𝑥1/2 (𝑥𝑥 + 3)
i. 4𝑥𝑥 −5/2 (𝑥𝑥 + 16) =
a-i: Use www.desmos.com to check graphs.
J
Area = 9/8 square units
4(𝑥𝑥+16)
𝑥𝑥 5/2
4. Lines: writing equations
1
a. 𝑦𝑦 = − 3 𝑥𝑥 −
7
9
b. 𝑦𝑦 = 8 𝑥𝑥 + 4
c. 𝑦𝑦 = 𝑥𝑥 + 2
1
d. 𝑦𝑦 = 5 𝑥𝑥 +
5
e. 𝑦𝑦 = 2 𝑥𝑥 +
3
11
3
26
5
27
2
f. 𝑦𝑦 = − 117 𝑥𝑥
5. Common graphs
(i) Power functions
Use www.desmos.com to check graphs. The pattern is that even power functions look like a “U,”
and the bend in the U gets flatter and the ends of the U get steeper as the exponent gets bigger.
Odd power functions look like snakes, and the bend in the snake gets flatter and the ends of the
snake get steeper as the exponent gets bigger.
(ii) Square root functions
Use www.desmos.com to check graphs. The pattern is that the original square root function (𝑦𝑦 =
𝑥𝑥1/2) looks like a half-rainbow, and as the numerator in the exponent increases the function starts
to bend upward. There are also domain issues to consider when the fraction exponent reduces.
(iii) Cube root functions
Use www.desmos.com to check graphs. The pattern is that the original cube root function (𝑦𝑦 =
𝑥𝑥1/3) looks like a horizontal snake, and as the numerator in the exponent increases the function
starts to bend downward less, and then starts to bend upward. When the numerator of the
exponent is even, the negative portion of the graph flips positive and creates a sharp point in the
graph.
6. Zeros (a.k.a. 𝒙𝒙-intercepts)
a. 𝑥𝑥 = −5, 0
5
e. 𝑥𝑥 = − 4 , 2
j. 𝑥𝑥 = 3�1/3
𝜋𝜋
m. 𝑥𝑥 = 4 +
𝑛𝑛𝑛𝑛
2
b. 𝑥𝑥 = −4, 4
c. 𝑥𝑥 = −√10, √10
f. 𝑥𝑥 = 5 + 3√2, 5 − 3√2
k. 𝑥𝑥 = − 1⁄3 , −2
, where n is any integer
1
g. 𝑥𝑥 = − 3
d. 𝑥𝑥 = 0, √7, −√7
h. none
l. 𝑥𝑥 = 𝑛𝑛𝑛𝑛, where n is any integer
i. 𝑥𝑥 = −5, 0
7. Vertical asymptotes
Find ALL vertical asymptotes of each function.
a. 𝑥𝑥 = 0
e. 𝑥𝑥 = −1, 1
b. 𝑥𝑥 = −3, 3
f. 𝑥𝑥 = 𝑛𝑛𝑛𝑛
5
c. 𝑥𝑥 = √3
𝜋𝜋
d. 𝑥𝑥 = 4 (Note: 𝑥𝑥 = −1 is NOT a vertical asymptote)
𝑛𝑛𝑛𝑛
𝜋𝜋
g. 𝑥𝑥 = 34 + 17
h. 𝑥𝑥 = 6 +
𝑛𝑛𝑛𝑛
3
8. Arrow notation
a. 𝑓𝑓(𝑥𝑥) → ∞
b. 𝑓𝑓(𝑥𝑥) → −∞
c. 𝑓𝑓(𝑥𝑥) → 0
d. 𝑓𝑓(𝑥𝑥) → 0
e. 𝑓𝑓(𝑥𝑥) → ∞
f. 𝑓𝑓(𝑥𝑥) → −∞
g. 𝑓𝑓(𝑥𝑥) → 0
h. 𝑓𝑓(𝑥𝑥) → 5/4
i.
j.
𝑓𝑓(𝑥𝑥) → ∞
𝑓𝑓(𝑥𝑥) → 0
k. 𝑓𝑓(𝑥𝑥) → ∞
9. Graphing with transformations
Use www.desmos.com to check graphs
10. Graphing with zeros
Use www.desmos.com to check graphs
11. Graphing with trig functions
Use www.desmos.com to check graphs. In desmos, if you click on the intercepts and max/mins, you
can see the coordinates of the point. This will help you check to make sure your points are correct.
12. Evaluating trig functions
a. 1
b. −1
c.
√3
3
d. √2
e. −
2
√3
or −
2√3
3
f. −
√3
3
g. 4
13. Graphing exponential and log functions
(i) Exponential functions (𝑏𝑏 > 1)
Use www.desmos.com to check graphs. The pattern is that all exponential functions have the same
basic shape, but as the base (𝑏𝑏) increases, the curve gets steeper.
(ii) Log functions (𝑏𝑏 > 1)
Use www.desmos.com to check graphs. The pattern is that all log functions have the same basic
shape, but as the base (𝑏𝑏) increases, the curve gets flatter.
Also, it should be noted that log functions and exponential functions are inverses of one another.
This means that the x and y values get swapped. For example, if (1,2) on an exponential function
then (2,1) will be on the corresponding log function.
(iii) Exponential functions (𝑏𝑏 < 1)
Use www.desmos.com to check graphs. The pattern is that all exponential functions with base less
than 1 are reflections of exponential functions with base greater than 1.
(iv) Log functions (𝑏𝑏 < 1)
Use www.desmos.com to check graphs. The pattern is that all log functions with base less than 1
are reflections of log functions with base greater than 1.
14. Simplifying exponential and log functions
a. 32
b. 5
c. 1
d. 1/2
e. 𝑥𝑥
f. 𝑥𝑥
g. 𝑥𝑥
h. 𝑥𝑥
15. Log rules
(i) a. 2 + 2 log 5 𝑥𝑥
𝑎𝑎3 𝑏𝑏 6
(iii) ln �
3
3
√𝑥𝑥⋅ √𝑦𝑦
�
𝑧𝑧 2
(ii) a. ln �
√𝑐𝑐
b.
1
2
log 2 (𝑥𝑥 − 5) − 5 log 2 (𝑥𝑥 + 1) − 4
1
b. log �𝑥𝑥 3 �
1
� = 3𝑥𝑥 + 6𝑦𝑦 − 3 𝑧𝑧
𝑥𝑥𝑥𝑥𝑥𝑥
c. ln � 𝑦𝑦𝑦𝑦 �
16. Trig Identities
a. 1
b. 1
c. 1
d. – tan2 𝑥𝑥
e. 1
f. 0
1
c. 1 + 2 ln 𝑥𝑥 + 4 ln 𝑧𝑧 − 3 ln 𝑦𝑦