APPLIED CALCULUS

practice problems

Dejan Živković, Timea Bezdan

Singidunum University, Belgrade

January 2024, Version 1.1

Contents

Contents 1

0 Review 2

1 Sets 4

2 Inequalities 5

3 Straight line equations 6

4 Functions 7

5 Domains and ranges of functions 8

6 Graphs of elementary functions 9

7 Composition of functions 10

8 Injective and inverse functions 11

9 Limit of sequences 12

10 Limits of functions 13

11 Continuous functions 15

12 Derivatives 16

13 Analysis of functions 19

14 Integrals 21

1
0 Review

1. What is the 2007-th digit after the decimal point in the decimal representation of the
1
number ?
7

2. Write the numbers 𝑥 = 0.3131313131… and 𝑦 = 0.273273273273… as fractions, i.e.,

express them as 𝑚
𝑛 specifying integers 𝑚 and 𝑛. (Hint: Notice that 100𝑥 = 𝑥 + 31.)

3. Is the number 𝑥 = 0.9999999999… , i.e., the number that in decimal representation

contains infinitely many only digits 9, equal to the number 1?

4. Expand the following expressions.

(a) (2𝑥 + 3)2 (b) (3𝑥 − 𝑦)2

(c) (𝑥 + 3𝑦)(𝑥 − 3𝑦) (d) (𝑥 + 3𝑦)(𝑥 + 4𝑦)
√ √ √
(e) (2 𝑥 − 3) (f) ( 𝑥 + 5) ( 𝑥 − 5)
2

5. Factorize the following expressions.

(a) 𝑥2 − 7𝑥 + 12 (b) 𝑥2 + 𝑥 − 6
(c) 𝑥2 + 8𝑥 + 16 (d) 9𝑥2 + 9𝑥 + 2
(e) 9𝑥2 − 6𝑥 + 1 (f) 5𝑥2 − 5
(g) 3𝑥2 − 18𝑥 + 27 (h) 2𝑥2 − 12𝑥 + 16

6. Simplify the following expressions.

𝑥2 − 𝑥 − 6 𝑥2 + 3𝑥 − 4
(a) (b)
𝑥2 − 7𝑥 + 12 2 − 𝑥 − 𝑥2
1 1
2𝑥 4𝑥2 + 4𝑥 −
(c) 2 ÷ (d) 𝑥+ℎ 𝑥
𝑥 −1 𝑥−1 ℎ

7. Solve for 𝑥 the following equations.

5𝑥 + 3 5𝑥 − 4
(a) 2(𝑥 + 4) = 7𝑥 + 2 (b) −5=
2 4
𝑥 𝑥
(c) (𝑎 + 𝑏)𝑥 + 𝑥2 = (𝑥 + 𝑏)2 (d) − =𝑐
𝑎 𝑏

2
 where 𝑎, 𝑏, 𝑐 are constants and 𝑎 ≠ 𝑏.

8. Solve the following equations.

(a) 4𝑥 − 4𝑥2 = 0 (b) 2 + 𝑥 − 3𝑥2 = 0

√
(c) 4𝑥 (𝑥 − 4) = 𝑥 − 15 (d) 𝑥2 + 2 2𝑥 + 2 = 0
√
(e) 𝑥2 + 2 2𝑥 + 3 = 0 (f) 𝑥3 − 7𝑥2 + 3𝑥 = 0

9. Find the value(s) for 𝑘 so that the equation 𝑥2 + 𝑘𝑥 + (𝑘 + 3) = 0 has only one
solution.

10. Find the positive number so that the sum of that number and its square is 210.

11. Solve the following equations.

(a) 2𝑥3 − 9𝑥2 − 8𝑥 + 15 = 0 (b) 𝑥3 − 2𝑥 + 1 = 0

(c) 2𝑥3 − 5𝑥2 + 2𝑥 − 15 = 0

12. Solve the following systems of equations.

2𝑥 + 3𝑦 = 7 𝑥 − 2𝑦 = 4
(a) (b)
3𝑥 + 5𝑦 = 11 𝑥2 + 𝑦 2 = 5

3
1 Sets

1. Given the sets 𝐴 = {𝑥 ∈ 𝑅 ∶ 𝑥 > 0} and 𝐵 = {𝑦 ∈ 𝑅 ∶ 𝑦 > 0}, are 𝐴 and 𝐵 the same
sets?

2. Given the sets 𝐴 = {𝑥 ∈ 𝑈 ∶ 𝑥 ≤ 10}, 𝐵 = {𝑥 ∈ 𝑈 ∶ 𝑥 is a prime number}, and

𝐶 = {𝑥 ∈ 𝑈 ∶ 𝑥 is an even number}, where 𝑈 = {1, 2, 3, … , 19} is the universal set.
Find the following sets.

(a) 𝐴 ∩ 𝐵 (b) 𝐴 ∩ 𝐶 (c) 𝐵 ∩ 𝐶

(d) 𝐴 ∪ 𝐵 (e) 𝐴 ∪ 𝐶 (f) 𝐵 ∪ 𝐶
(g) 𝐴 ∪ 𝐵 ∪ 𝐶 (h) 𝐴 ∩ 𝐵 ∩ 𝐶 (i) (𝐴 ∪ 𝐵) ∩ 𝐶
(j) (𝐴 ∩ 𝐵) ∪ 𝐶 (k) 𝐴 ∩ 𝐵 (l) 𝐴 ∩ 𝐵

3. Let 𝐴, 𝐵, and 𝐶 be subsets of a universal set 𝑈. For each of the following equalities
determine whether it is true or not.

(a) 𝐴 − 𝐵 = 𝐴 ∩ 𝐵 (b) (𝐴 ∪ 𝐵) ∩ 𝐶 = 𝐴 ∪ (𝐵 ∩ 𝐶)
(c) (𝐴 ∪ 𝐵) ∩ 𝐵 = 𝐵 − 𝐴

4. List the elements of the following sets.

(a) {𝑥 ∈ 𝑅 ∶ 𝑥2 = 2} (b) {𝑥 ∈ 𝑅 ∶ 𝑥 ≥ 0 and 𝑥2 = 2}

√
(c) {𝑥 ∈ 𝑄 ∶ 𝑥2 = 2} (d) {({1} ∪ {2, 3}) ∩ (0, 2 2)}

5. Given the sets 𝐴 = [1, 5], 𝐵 = [3, 9), 𝐶 = {1, 5}, and 𝐷 = [5, ∞), find the following
sets.

(a) 𝐴 ∩ 𝐵 (b) 𝐴 ∪ 𝐵 (c) 𝐴 − 𝐶

(d) 𝐵 ∩ 𝐶 (e) 𝐶 − 𝐵 (f) 𝐵 − 𝐶
(g) 𝐵 − (𝐵 − 𝐶) (h) 𝐴 ∪ 𝐷 (i) 𝐶 ∩ 𝐷

6. Let 𝐴 and 𝐵 be intervals on the real line. Is it always the case that 𝐴 ∩ 𝐵 is an
interval too? What about 𝐴 ∪ 𝐵 being an interval?

4
2 Inequalities

1. Solve the following inequalities.

1−𝑥 3𝑥 − 7 √
(a) ≥ (b) 2(3 − 𝑥) ≤ 3(1 − 𝑥)
2 3
3𝑥 2𝑥
(c) +3<0 (d) >1
1−𝑥 2𝑥 + 3

2. Solve the following inequalities.

(a) 2𝑥 − 3 ≥ 4 + 7𝑥 (b) 8(𝑥 + 1) − 2 < 5(𝑥 − 6) + 7
2
𝑥 − 3𝑥 + 7
(c) <1 (d) (2𝑥 + 7)(5 − 11𝑥) ≤ 0
𝑥2 + 1
(e) 𝑥2 − 2𝑥 − 3 < 0 (f) 2𝑥2 − 3𝑥 > 4
(g) 2𝑥2 − 3𝑥 < −4 (h) −𝑥2 − 𝑥 + 2 > 0
(i) 𝑥2 + 𝑥 − 12 ≥ 0 (j) −𝑥2 − 2𝑥 + 3 ≤ 0
2𝑥 + 3
(k) 𝑥2 + 2𝑥 − 15 > 0 (l) ≥0
𝑥−4
2𝑥 + 3 4−𝑥
(m) <1 (n) 2 ≤0
𝑥−4 𝑥 + 2𝑥 − 3
𝑥+1 𝑥2 − 1
(o) ≥0 (p) 2 ≤0
𝑥−2 𝑥 + 2𝑥
−𝑥2 + 4 𝑥3 − 1
(q) 3 ≥0 (r) ≤0
𝑥 + 3𝑥 𝑥4 − 81

3. Solve the following inequalities.

(a) (𝑥 − 4)(9 − 5𝑥)(2𝑥 + 3) < 0 (b) (𝑥 − 3)(2𝑥 + 1)2 ≤ 0
(c) 𝑥3 − 2𝑥2 − 5𝑥 + 6 < 0 (d) −2𝑥3 + 𝑥2 + 15𝑥 − 18 ≤ 0
(e) 𝑥3 − 𝑥2 − 5𝑥 − 3 > 0 (f) 𝑥3 + 3𝑥2 + 5𝑥 + 3 ≤ 0
(g) 𝑥3 + 3𝑥2 − 4𝑥 − 12 ≤ 0 (h) 9𝑥3 − 12𝑥2 − 3𝑥 + 4 > 0
(i) 𝑥3 + 3𝑥2 + 7𝑥 + 21 ≥ 0 (j) 𝑥4 + 2𝑥3 − 13𝑥2 − 14𝑥 + 24 > 0
(k) 6𝑥4 + 𝑥3 − 15𝑥2 ≤ 0

4. Write the following sets of real numbers as (union of) intervals.

(a) 𝐴 = {𝑥 ∈ 𝑅 ∶ 𝑥2 − 3𝑥 + 2 ≤ 0} (b) 𝐵 = {𝑥 ∈ 𝑅 ∶ 𝑥2 − 3𝑥 + 2 ≥ 0}
(c) 𝐶 = {𝑥 ∈ 𝑅 ∶ 𝑥2 − 3𝑥 > 3} (d) 𝐷 = {𝑥 ∈ 𝑅 ∶ 𝑥2 − 5 > 2𝑥}
(e) 𝐸 = {𝑡 ∈ 𝑅 ∶ 𝑡2 − 3𝑡 + 2 ≤ 0} (f) 𝐹 = {𝛼 ∈ 𝑅 ∶ 𝛼2 − 3𝛼 + 2 ≥ 0}

5
3 Straight line equations

1. Find the equations of the following lines in explicit (slope-intercept) and general
forms.

(a) Line passes through the origin and the point (−2, 3).
(b) Line’s slope is 2 and passes through the point (5, −1).
(c) Line’s slope is −3 and its 𝑦-intercept is (0, 7).
(d) Line passes through the point (−3, 2) and it is parallel to the line 2𝑥 − 𝑦 − 3 = 0.
(e) Line passes through the point (1, 4) and it is perpendicular to the line 𝑥+3𝑦 = 0.
(f) Line passes through the point (1, −1) and it is perpendicular to the 𝑦-axes.

2. Let ℓ1 and ℓ2 be two lines such that ℓ1 passes through the points 𝐴(1, 3) and 𝐵(2, 5),
while ℓ2 passes through the point 𝐶(0, 2) and its slope is − 23 .

(a) Find the equations of the line ℓ1 in explicit (slope-intercept) and general forms.
(b) Find the equations of the line ℓ2 in explicit (slope-intercept) and general forms.
(c) Find the coordinates of the point of intersection of the lines ℓ1 and ℓ2 .

3. For any real number 𝑎, let ℓ𝑎 be the line given by the equation 𝑦 = 𝑎𝑥 + 𝑎2 .

(a) Draw the lines ℓ𝑎 for 𝑎 = −2, −1, − 12 , 0, 12 , 1, 2.

(b) Does the point (3, 2) lie on some line(s) ℓ𝑎 ? (Note: 𝑎 can be any real number,
not only one of those from part (a).) In other words, for what values of 𝑎 does
the point (3, 2) lie on ℓ𝑎 ?
(c) What points in the plane lie on at least one of the lines ℓ𝑎 ?

4. Given the sets 𝐶 = {(𝑥, 𝑦) ∈ 𝑅2 ∶ 𝑥2 + 𝑦2 = 5}, 𝐸 = {(𝑥, 𝑦) ∈ 𝑅2 ∶ 𝑥2 + 2𝑦2 = 6},

and 𝐿 = {(𝑥, 𝑦) ∈ 𝑅2 ∶ 2𝑥 + 𝑦 − 3 = 0}, find the following sets.

(a) 𝐿 ∩ 𝐶 (b) 𝐿 ∩ 𝐸 (c) 𝐶 ∩ 𝐸

5. Given the sets 𝐶 = {(𝑥, 𝑦) ∈ 𝑅2 ∶ 𝑥2 + 𝑦2 = 1} and 𝐿 = {(𝑥, 𝑦) ∈ 𝑅2 ∶ 𝑎𝑥 + 𝑦 = 2},

where 𝑎 is a constant, find the values for 𝑎 such that the set 𝐶 ∩ 𝐿 contains only one
element.

6
4 Functions

𝑥−5
1. Given the function 𝑓(𝑥) = , calculate the following expressions.
𝑥2 + 4

(a) 𝑓(2) (b) 𝑓(3.5) (c) 𝑓(𝑎 + 1)

√
(d) 𝑓( 𝑎) (e) 𝑓(𝑎 ) 2
(f) 𝑓(𝑎) + 𝑓(1)

𝑥 √
2. Given the functions 𝑓(𝑥) = and 𝑔(𝑥) = 𝑥 − 1, calculate the following expres-
𝑥+1
sions.
𝑓(3)
(a) 𝑓(1) + 𝑔(1) (b) 𝑓(2)𝑔(2) (c)
𝑔(3)
(d) 𝑓(𝑎 − 1) + 𝑔(𝑎 + 1) (e) 𝑓(𝑎2 + 1)𝑔(𝑎2 + 1) (f) 𝑓(1)𝑔2 (3)

3. Given the function 𝑓(𝑥) = 𝑥2 −3𝑥+4, calculate and simplify the following expressions.
𝑓(1 + ℎ) − 𝑓(1) 𝑓(𝑎 + ℎ) − 𝑓(𝑎)
(a) 𝑓(𝑎 + 𝑏) (b) (c)
ℎ ℎ

4. Given the function 𝑓(𝑥) satisfying 𝑓 (2𝑥 + 3) = 𝑥2 for all real numbers 𝑥, calculate
the following expressions.

(a) 𝑓(0) (b) 𝑓(3)

(c) 𝑓(𝑥) (d) 𝑓(𝑓(2))

5. Given the function 𝑓(𝑥) satisfying 𝑓 ( 𝑥+1

1
) = 2𝑥 − 12 for all real numbers 𝑥 ≠ −1,
calculate the following expressions.

(a) 𝑓(1) (b) 𝑓(0)

(c) 𝑓(𝑥) (d) 𝑓(𝑡)
(e) 𝑓(𝑓(2)) (f) 𝑓(𝑓(1/2) + 10)

7
5 Domains and ranges of functions

1. For each of the following functions find its domain.

2
(a) 𝑓(𝑥) = 𝑥2 − 5 (b) 𝑓(𝑥) =
5𝑥 + 6
1 1
(c) 𝑓(𝑥) = (d) 𝑓(𝑥) =
𝑥2 − 5 𝑥2 − 2𝑥 − 3
1 1 √
(e) 𝑓(𝑥) = √ (f) 𝑓(𝑥) = − 𝑥+3
2𝑥 − 3 1 − 2𝑥
3 √ 1
(g) 𝑓(𝑥) = 2
+ 2𝑥 + 5 (h) 𝑓(𝑥) = √
1−𝑥 𝑥2 + 3𝑥 − 10

2. For each of the following functions find its range.

(a) 𝑓(𝑥) = 𝑥2 − 5 (b) 𝑓(𝑥) = 𝑥2 − 2𝑥 − 3

2 1
(c) 𝑓(𝑥) = (d) 𝑓(𝑥) = 3 −
5𝑥 + 6 2𝑥 − 1
1 1
(e) 𝑓(𝑥) = √ (f) 𝑓(𝑥) =
2𝑥 − 3 𝑥2 − 5
1
(g) 𝑓(𝑥) =
𝑥2 − 2𝑥 − 3

3. Given the function 𝑓(𝑥) satisfying 𝑓(2𝑥 + 3) = 𝑥2 for all real numbers 𝑥, find its
domain and range.

1
4. Given the function 𝑓(𝑥) satisfying 𝑓( ) = 2𝑥 − 12 for all real numbers 𝑥, find its
𝑥+1
domain and range.

5. For each of the following functions find its domain and range.

(a) 𝑓(𝑥) = 2𝑥2 + 3 (b) 𝑔(𝑥) = −2𝑥2 + 4𝑥

1
(c) ℎ(𝑥) = 4𝑥 + 𝑥2 (d) 𝑚(𝑥) =
1 + 𝑥2
1
(e) 𝑛(𝑥) =
3 + 2𝑥 + 𝑥2

8
6 Graphs of elementary functions

3
1. Consider the functions 𝑓(𝑥) = −2𝑥2 + 4𝑥 − and 𝑔(𝑥) = 2𝑥 − 1.
2
(a) Sketch the graphs of 𝑓(𝑥) and 𝑔(𝑥) on the same diagram of the Cartesian plane.
(b) Find the coordinates of the point(s) where the graph of 𝑓(𝑥) intersects the 𝑥-axes
and 𝑦-axes, as well as the coordinates of the vertex point of the graph of 𝑓(𝑥).
(c) Find the coordinates of the point(s) of intersection of the graphs of 𝑓(𝑥) and
𝑔(𝑥).

1
2. Consider the functions 𝑓(𝑥) = 𝑚𝑥 + 𝑛 and 𝑔(𝑥) = .
𝑥
(a) For what values of 𝑚 and 𝑛 does the graph of 𝑓(𝑥) pass through the point (−1, 1)
and intersect the graph of 𝑔(𝑥) in exactly one point?
(b) For what values of 𝑚 and 𝑛 does the graph of 𝑓(𝑥) not intersect the graph of
𝑔(𝑥) at all?

3. Sketch the graph of the following functions.

(a) 𝑓(𝑥) = 𝑥2 + 2𝑥 − 3 (b) 𝑓(𝑥) = −(𝑥 − 3)2 + 2

(c) 𝑓(𝑥) = |𝑥 + 1| (d) 𝑓(𝑥) = |2𝑥 − 1| + 5
√
(e) 𝑓(𝑥) = 1 − 𝑥2

9
7 Composition of functions

1. Given the following functions 𝑓(𝑥) = 𝑥2 + 1 and 𝑔(𝑥) = 𝑥 + 1, calculate the following
expressions.

(a) (𝑓 ∘ 𝑔)(1) (b) (𝑔 ∘ 𝑓)(1) (c) (𝑓 ∘ 𝑔)(𝑥)

√
(d) (𝑔 ∘ 𝑓)(𝑥) (e) (𝑓 ∘ 𝑔)(𝑎2 ) (f) (𝑔 ∘ 𝑓)( 𝑎)

2. Specify the functions 𝑓(𝑥) and 𝑔(𝑥) so that 𝑔(𝑥) has the form 𝑥𝑟 with 𝑟 ≠ 1, and the
composition (𝑔 ∘ 𝑓)(𝑥) is equal to the following expressions.
√ 1
(a) 𝑥2 + 1 (b)
𝑥+1

10
8 Injective and inverse functions

1. For each of the following functions determine whether it is injective (1–1).

√
(a) 𝑓(𝑥) = 5 (b) 𝑓(𝑥) = 𝑥
(c) 𝑓(𝑥) = 𝑥2 − 5 (d) 𝑓(𝑥) = 𝑥3 + 2𝑥
(e) 𝑓(𝑥) = 1
1+𝑥2 (f) 𝑓(𝑥) = ln(𝑥 + 1)

2. For each of the following functions find its inverse function.

(a) 𝑓(𝑥) = 3𝑥 − 2 (b) 𝑓(𝑥) = 𝑥5 + 3

√
(c) 𝑓(𝑥) = 1 + 2𝑥 7 (d) 𝑓(𝑥) = 2𝑥3 − 1
1 3

11
9 Limit of sequences

1. For each of the following sequences find its limit, if it exists.

1 5
(a) lim √ (b) lim (7 − )
𝑛→∞ 𝑛 𝑛→∞ 𝑛2
3𝑛2 − 4000 𝑛2 − 12345
(c) lim (d) lim
𝑛→∞ 2𝑛2 + 10000 𝑛→∞ 𝑛+1
5𝑛2 + 4 1
(e) lim (f) lim ( + (−1)𝑛 )
𝑛→∞ 2𝑛3 + 3 𝑛→∞ 𝑛

2. Suppose $50,000 is deposited in a bank account and the annual interest rate is 2%.

(a) What amount will the account have after one year if the interest is compounded
quarterly and what if it is compounded monthly?
(b) If the interest is compounded 𝑛 times a year, express the amount 𝐴𝑛 after one
year in terms of 𝑛.
(c) Does lim 𝐴𝑛 exist?
𝑛→∞

12
10 Limits of functions

1. Calculate the following limits, if they exist.

1
(a) lim √ (b) lim (15 − 16𝑥−3 ) (c) lim 5−𝑥
𝑥→+∞ 𝑥−1 𝑥→+∞ 𝑥→+∞
√ 𝑥2 + 9 𝑥2 + 9
(d) lim 𝑥 (e) lim (f) lim
𝑥→+∞ 𝑥→+∞ 𝑥3 + 1 𝑥→+∞ 𝑥2 + 1
𝑥2 + 9 3𝑥2 + 3 𝑥
(g) lim (h) lim 2 + 7𝑥 − 39
(i) lim 3
𝑥→+∞ 𝑥 + 1 𝑥→+∞ 5𝑥 𝑥→+∞ 𝑥 +5
(2𝑥 + 1)4 |𝑥| |𝑥|
(j) lim (k) lim (l) lim
𝑥→+∞ (3𝑥2 + 1)2 𝑥→+∞ 𝑥 𝑥→−∞ 𝑥

(m) lim 𝑥 sin 𝑥

𝑥→−∞

2. The concentration 𝐶 of a drug in the patient’s bloodstream 𝑡 hours after it was in-
jected is given by the function

0.15𝑡
𝐶(𝑡) = .
𝑡2 + 3

(a) Find lim 𝐶(𝑡).

𝑡→+∞

(b) Interpret the result in (a).

3. Calculate the following one-sided limits, if they exist.

√ 1
(a) lim (2𝑥 + 5) (b) lim 𝑥 − 1 (c) lim √
𝑥→7− 𝑥→1+ 𝑥→0+ 𝑥
1 1
(d) lim (e) lim sin (f) lim (2 − 3 𝑥 )
1

𝑥3 𝑥
𝑥→0− 𝑥→0− 𝑥→0+

(g) lim (2 − 3 )
1
𝑥
𝑥→0−

4. Calculate the following limits, if they exist.

(a) lim (2𝑥 + 5) (b) lim (𝑥2 + 3𝑥 − 4) (c) lim (𝑥2 − 5𝑥 − 8)3
𝑥→−7 𝑥→2 𝑥→7
√
3𝑥 − 5 2 𝑥 𝑥2 + 1
(d) lim (𝑥 + 3) 2007
(e) lim ( ) (f) lim
𝑥→−4 𝑥→0 2𝑥 + 7 𝑥→1 𝑥+1

13
 𝑥2 − 4 𝑥2 − 4
(g) lim 25+4𝑥 (h) lim (i) lim
𝑥→−2 𝑥→−2 𝑥2+𝑥−2 𝑥→−2 𝑥2−𝑥+2
𝑥2 − 6𝑥 𝑥2 + 25 𝑥−3
(j) lim 2 (k) lim (l) lim 2
𝑥→6 𝑥 − 5𝑥 − 6 𝑥→5 𝑥 − 5 𝑥→3 𝑥 + 9
√
𝑥2 + 𝑥 − 2 𝑥2 + 𝑥 − 2 𝑥−2
(m) lim 2
(n) lim 2
(o) lim
𝑥→1 𝑥 − 1 𝑥→−1 𝑥 − 1 𝑥→4 𝑥 − 4
√ 1 1
𝑥−5 𝑥−3 −
(p) lim √ (q) lim (r) lim 𝑥 2
.
𝑥→2 2 − 𝑥 − 1 𝑥→9 𝑥 − 9 𝑥→2 𝑥 − 2
1 1
√ − √
𝜋
(s) lim (t) lim sin( )
𝑥 2
𝑥→2 𝑥−2 𝑥→0 𝑥

𝑓(𝑥 + ℎ) − 𝑓(𝑥)
5. For each of the following functions find lim .
ℎ→0 ℎ

(a) 𝑓(𝑥) = 1 (b) 𝑓(𝑥) = 4𝑥 − 13 (c) 𝑓(𝑥) = 𝑥2 + 3

1 √
(d) 𝑓(𝑥) = 𝑥3 (e) 𝑓(𝑥) = (f) 𝑓(𝑥) = 𝑥
𝑥

14
11 Continuous functions

⎧ 𝑥2 ,𝑥<1
{
1. Given the function 𝑓(𝑥) = 1 ,1≤𝑥<2 ,
⎨1
{ ,𝑥≥2
⎩𝑥

(a) sketch the graph of 𝑓(𝑥) for 𝑥 ∈ [0, 5].

(b) find the points where 𝑓(𝑥) is not continuous.

𝑥2 + 𝑥 − 2
2. Given the function 𝑓(𝑥) = √ ,
1− 𝑥

(a) find the domain of 𝑓(𝑥).

(b) find lim 𝑓(𝑥).
𝑥→1
(c) is it possible to define 𝑓(1) so that 𝑓(𝑥) is continuous at 1?

1
3. Given the function 𝑓(𝑥) = sin ,
𝑥

(a) find the domain of 𝑓(𝑥).

(b) find lim 𝑓(𝑥).
𝑥→0
(c) is it possible to define 𝑓(0) so that 𝑓(𝑥) is continuous at 0?

4. Given the function

⎧ 𝑥3 , 𝑥 < −1
{
𝑓(𝑥) = 𝑎𝑥 + 𝑏
⎨ , −1≤𝑥<1 ,
{ 𝑥2 + 2 ,𝑥≥1
⎩
where 𝑎 and 𝑏 are constants, find 𝑎 and 𝑏 so that 𝑓(𝑥) is continuous function.

5. Find constant 𝑘 so that the function

3𝑥 + 2 ,𝑥<2
𝑓(𝑥) = {
𝑥2 + 𝑘 ,𝑥≥2

is continuous. (Hint: Consider the one-sided limits.)

15
12 Derivatives

1. For each of the following functions 𝑓(𝑥) find its derivative 𝑓 ′ (𝑥) by definition.

(a) 𝑓(𝑥) = 2𝑥2 + 1 (b) 𝑓(𝑥) = 𝑥3 − 3𝑥

1
(c) 𝑓(𝑥) = 𝑥4 (d) 𝑓(𝑥) =
𝑥2

2. For each of the following functions 𝑦 = 𝑓(𝑥) find 𝑦′ .

(a) 𝑦 = −𝜋 (b) 𝑦 = 2𝑥9 + 3𝑥

(c) 𝑦 = 𝑥2 + 5𝑥 − 7 (d) 𝑦 = 𝑥 (𝑥 − 1)
(e) 𝑦 = (2𝑥 − 3)(5 − 6𝑥) (f) 𝑦 = (𝑥2 + 5)3
23 𝑥−1
(g) 𝑦 = (h) 𝑦 =
𝑥4 𝑥
𝑥−1 √ √
(i) 𝑦 = (j) 𝑦 = 𝑥 ( 𝑥 + 1)
𝑥+1

3. For each of the following functions 𝑓(𝑥) find 𝑓 ′ (𝑎) for given 𝑎.
2 4
(a) 𝑓(𝑥) = 𝑥3 − 4𝑥, 𝑎 = 1 (b) 𝑓(𝑥) = + , 𝑎=2
𝑥3𝑥
√
(c) 𝑓(𝑥) = 3𝑥 3 − 5𝑥 3 , (d) 𝑓(𝑥) = 𝜋𝑥 − 2 𝑥, 𝑎 = 4
1 2
2
𝑎 = 27
𝑥2 + 1
(e) 𝑓(𝑥) = (𝑥2 + 3)(𝑥3 + 2), 𝑎=1 (f) 𝑓(𝑥) = , 𝑎=2
2𝑥 − 3
𝑥2 + 3𝑥 − 5
(g) 𝑓(𝑥) = , 𝑎=1
𝑥2 − 7𝑥 + 5

4. Consider the function 𝑓(𝑥) = 3𝑥4 − 6𝑥2 + 2.

(a) Find the slope of 𝑓(𝑥) at the point with 𝑥-coordinate equal to 2.
(b) Find the tangent line of 𝑓(𝑥) at the point (1, −1).
(c) Find the point(s) on the graph of the function where the tangent line of 𝑓(𝑥) is
horizontal.

5. For each of the following functions 𝑦 = 𝑓(𝑥) find 𝑦′ .

16
 (a) 𝑦 = 5 cos 𝑥 − 2𝑥 (b) 𝑦 = 1 − 2 tan 𝑥
(c) 𝑦 = sin 𝑥 − 𝑥 2
(d) 𝑦 = 𝑥2 sin 𝑥
1
(e) 𝑦 = cos2 𝑥 (f) 𝑦 =
cos 𝑥
cos 𝑥
(g) 𝑦 = sin 𝑥 ⋅ cos 𝑥 (h) 𝑦 =
𝑥3 + 1
(i) 𝑦 = (𝑥 + cos 𝑥)2 (j) 𝑦 = (sin 𝑥 + cos 𝑥)2

6. Given 𝑦 = sin𝑛 𝑥 for 𝑛 = 1, 2, 3, …,

(a) find 𝑦′ for 𝑛 = 2, 3, and 4.
(b) find the general formula for 𝑦′ in terms of 𝑛.

7. For each of the following functions 𝑦 = 𝑓(𝑥) find 𝑦′ .

(a) 𝑦 = 2𝑥3 − 4𝑒𝑥 − 5 (b) 𝑦 = ln 𝑥 − 1
(c) 𝑦 = 𝑒 + ln 𝑥
𝑥
(d) 𝑦 = 𝑥2 + ln 𝑥2
√ √
(e) 𝑦 = 𝑒𝑥 + 𝑥 (f) 𝑦 = ln 𝑥 − 1
(g) 𝑦 = 𝑒𝑥 sin 𝑥 (h) 𝑦 = cos 𝑥 ln 𝑥
(i) 𝑦 = (𝑥 + 1)𝑒
2 𝑥
(j) 𝑦 = (𝑥2 + 1) ln 𝑥
𝑒𝑥 ln 𝑥
(k) 𝑦 = (l) 𝑦 =
cos 𝑥 sin 𝑥
𝑥3 + 3𝑥2 + 6𝑥 − 2 𝑥3 + 3𝑥2 + 6𝑥 − 2
(m) 𝑦 = (n) 𝑦 =
𝑒𝑥 ln 𝑥

8. For each of the following functions 𝑓(𝑥) find 𝑓 ′ (𝑎) for given 𝑎.
ln 𝑥
(a) 𝑓(𝑥) = 𝑒𝑥 tan 𝑥, 𝑎=0 (b) 𝑓(𝑥) = , 𝑎=1
𝑥2 + 1

9. For each of the following functions 𝑓(𝑥) find 𝑓 ′ (𝑥).

𝑥−2
(a) 𝑓(𝑥) = 𝑥 + 1 + (𝑥 + 1)2 (b) 𝑓(𝑥) =
𝑥4 + 1
−1 √
1
(c) 𝑓(𝑥) = ( ) (d) 𝑓(𝑥) = 1 − 𝑥2
1+𝑥
𝑎𝑥 + 𝑏 1
(e) 𝑓(𝑥) = (f) 𝑓(𝑥) =
𝑐𝑥 + 𝑑 (1 + 𝑥2 )2
𝑥 1−𝑥
(g) 𝑓(𝑥) = √ (h) 𝑓(𝑥) = √
1+ 𝑥 1+𝑥
√
(i) 𝑓(𝑥) = √𝑥 + 𝑥
3

17
10. For each of the following functions 𝑦 = 𝑓(𝑥) find 𝑦′ .
√
(a) 𝑦 = (2𝑥3 + 5𝑥2 )6 (b) 𝑦 = 9 + 4𝑥
√
4𝑥2 + 5
(c) 𝑦 = (d) 𝑦 = cos 5𝑥
3
(e) 𝑦 = sin(6𝑥 − 7) (f) 𝑦 = sin4 𝑥
(g) 𝑦 = cos5 (6𝑥 − 7) (h) 𝑦 = 4𝑥 sin 3𝑥
tan 𝑥
(i) 𝑦 = tan(8𝑥3 + 1) (j) 𝑦 =
𝑥+2
(k) 𝑦 = 2𝑒3𝑥 + 4𝑥 − 5 (l) 𝑦 = 𝑥 𝑒𝑥
2

𝑥2
(m) 𝑦 = (n) 𝑦 = ln 8𝑥
𝑒𝑥
(o) 𝑦 = ln(5 − 2𝑥) (p) 𝑦 = ln(1 − 𝑥2 )
√
(q) 𝑦 = ln 2𝑥 + 11 (r) 𝑦 = 3𝑥 ln 𝑥
(s) 𝑦 = ln(ln 𝑥) (t) 𝑦 = 𝑒𝑥 ln 𝑥
2

(u) 𝑦 = 𝑒tan 𝑥 (v) 𝑦 = tan(𝑒𝑥 )

(w) 𝑦 = sin(𝑒5𝑥 ) (x) 𝑦 = 𝑒sin 5𝑥
(y) 𝑦 = cos[ln(4𝑥2 + 9)] (z) 𝑦 = ln[cos(4𝑥2 + 9)]

11. For each of the following functions 𝑦 = 𝑓(𝑥) find 𝑦″ .

1
(a) 𝑦 = 𝑥2 + 𝑥 − 1 − 𝑒𝑥 (b) 𝑦 = 1 − ln 𝑥 −
𝑥
4
(c) 𝑦 = (𝑥 + 1)4 (d) 𝑦 = (𝑥2 + 1)
√ √
(e) 𝑦 = 𝑥 − 2 (f) 𝑦 = 3 𝑥 − 1

𝑥2 𝑥3 𝑥4 𝑥5 𝑥6
12. Find 𝑓 ′ (𝑥), 𝑓 ″ (𝑥), and 𝑓 (3) (𝑥) if 𝑓(𝑥) = 1 + 𝑥 + + + + + .
2 6 24 120 720

18
13 Analysis of functions

1. For each of the following functions 𝑓(𝑥) find its intervals of monotonicity (i.e., where
the function is increasing and decreasing).

(a) 𝑓(𝑥) = 2𝑥2 − 5𝑥 + 6 (b) 𝑓(𝑥) = 1 + 3𝑥 − 𝑥3

(c) 𝑓(𝑥) = 𝑥3 + 6𝑥2 − 63𝑥 (d) 𝑓(𝑥) = 2𝑥3 + 9𝑥2 − 6𝑥 + 7
4
(e) 𝑓(𝑥) = 3𝑥4 + 4𝑥3 − 24𝑥2 − 48𝑥 (f) 𝑓(𝑥) = 𝑥 +
𝑥

2. For each of the following functions 𝑓(𝑥) find the type of its critical points (i.e., whether
the point is maximum or minimum).

(a) 𝑓(𝑥) = −𝑥2 + 7𝑥 − 13 (b) 𝑓(𝑥) = 𝑥4 − 2𝑥3

𝑥2 + 𝑥 + 1
(c) 𝑓(𝑥) = 𝑥5 − 15𝑥3 (d) 𝑓(𝑥) =
𝑥+1
𝑥 𝑥2
(e) 𝑓(𝑥) = (f) 𝑓(𝑥) =
1 + 𝑥2 𝑥3 − 𝑥

3. For each of the following functions 𝑓(𝑥) find the intervals where the function is convex
and concave.
√
(a) 𝑓(𝑥) = 𝑥 (b) 𝑓(𝑥) = 𝑥3 − 6𝑥2 + 9𝑥
2
(c) 𝑓(𝑥) = 3𝑥5 − 9𝑥4 + 8𝑥3 (d) 𝑓(𝑥) = 𝑥 +
𝑥

4. For each of the following functions 𝑓(𝑥) find its inflection points.
1 1
(a) 𝑓(𝑥) = 2𝑥3 + 9𝑥2 − 108𝑥 + 35 (b) 𝑓(𝑥) = 1 − +
𝑥 𝑥2

5. For each of the following functions 𝑓(𝑥) find its domain, asymptotes, 𝑥- and 𝑦-
intercepts, intervals of monotonicity, intervals of convexity/concavity, and sketch the
graph of the function.
𝑥 1 + 𝑥2
(a) 𝑓(𝑥) = (b) 𝑓(𝑥) =
1 + 𝑥2 1+𝑥
3𝑥(1 − 𝑥) 𝑥3
(c) 𝑓(𝑥) = (d) 𝑓(𝑥) =
1 + 𝑥2 4 − 𝑥2
2𝑥 − 1
(e) 𝑓(𝑥) = (f) 𝑓(𝑥) = √1 + 𝑥2
(𝑥 − 1)2

19
 √
(g) 𝑓(𝑥) = 1 − 𝑥2 (h) 𝑓(𝑥) = 𝑥2 𝑒−𝑥
𝑒𝑥 𝑒𝑥 − 𝑒−𝑥
(i) 𝑓(𝑥) = (j) 𝑓(𝑥) =
1 + 𝑒𝑥 2
1+𝑥
(k) 𝑓(𝑥) = ln (1 + 𝑥2 ) (l) 𝑓(𝑥) = ln √
1−𝑥

20
14 Integrals

1. Find the following indefinite integrals.

4
(a) ∫ 2𝑥5 𝑑𝑥 (b) ∫ (3 − √ ) 𝑑𝑥
𝑥
√
(c) ∫(𝑥7 − 3𝑥 + 2) 𝑑𝑥 (d) ∫(𝑥2 − 𝑥 + 3) 𝑑𝑥

3
(e) ∫(6𝑥5 − 2𝑥−4 − 7𝑥) 𝑑𝑥 (f) ∫ ( − 5 + 4𝑒𝑥 + 7𝑥 ) 𝑑𝑥
𝑥
1 1 𝑥−2
(g) ∫ ( 2
− 4 ) 𝑑𝑥 (h) ∫ √ 𝑑𝑥
𝑥 𝑥 3𝑥 𝑥
2
𝑥2 + 1 √ 1
(i) ∫ √ 𝑑𝑥 (j) ∫ ( 𝑥 + √ ) 𝑑𝑥
𝑥 𝑥
𝑥4 − 1 𝑥2 + 1
(k) ∫ 𝑑𝑥 (l) ∫ 𝑑𝑥
𝑥2 + 1 𝑥2

(m) ∫(𝑥2 − 5𝑥 + 1)(2 − 3𝑥) 𝑑𝑥 (n) ∫(𝑥2 − 3)2 𝑑𝑥

tan 𝑥
(o) ∫(cos 𝑥 + 2 sin 𝑥) 𝑑𝑥 (p) ∫ 𝑑𝑥
cos 𝑥

2. Find the following indefinite integrals.

(a) ∫ 2𝑥(𝑥2 + 1)9 𝑑𝑥 (b) ∫ 𝑥4 √𝑥5 + 6 𝑑𝑥

(c) ∫ 𝑥 sin 𝑥2 𝑑𝑥 (d) ∫ sin 𝑥 cos2 𝑥 𝑑𝑥

1
𝑒𝑥
(e) ∫ 2𝑥𝑒𝑥 𝑑𝑥 (f) ∫
2
𝑑𝑥
𝑥2

(g) ∫ 𝑥𝑒−𝑥 (h) ∫ 𝑥2 𝑒𝑥

2 3
+1 −1
𝑑𝑥 𝑑𝑥

𝑥 sin 𝑥1
(i) ∫ 𝑑𝑥 (j) ∫ 𝑑𝑥
𝑥2 + 1 𝑥2
𝜋+𝑥
(k) ∫ cos (𝑥 + 𝜋3 ) 𝑑𝑥 (l) ∫ sin ( ) 𝑑𝑥
5
sin 2𝑥 sin 2𝑥
(m) ∫ 𝑑𝑥 (n) ∫ 𝑑𝑥
1 + cos2 𝑥 1 + sin 𝑥
𝑥3 + 𝑥
(o) ∫(𝑥 + 1)(𝑥2 + 2𝑥 + 3)7 𝑑𝑥 (p) ∫ 𝑑𝑥
(𝑥4 + 2𝑥2 + 3)11

21
 √
𝑒 𝑥
(q) ∫(𝑒 − 3𝑥) (𝑒 − 3) 𝑑𝑥
𝑥 4 𝑥
(r) ∫ √ 𝑑𝑥
𝑥
ln(𝑥 + 1) 1
(s) ∫ 𝑑𝑥 (t) ∫ 𝑑𝑥
𝑥+1 𝑥 ln 𝑥
1
(u) ∫ 𝑑𝑥 (v) ∫(𝑥 + 1)15 𝑑𝑥
2𝑥 + 7
𝑥
(w) ∫ √ 𝑑𝑥 (x) ∫ 𝑥(𝑥 + 1)15 𝑑𝑥
𝑥+1
1
(𝑥2 − 1)𝑒𝑥+ 𝑥
(y) ∫ 𝑑𝑥
𝑥2

3. Find the following indefinite integrals.

𝑥
(a) ∫ 𝑥𝑒−𝑥 𝑑𝑥 (b) ∫ 𝑑𝑥
sin2 𝑥
(c) ∫ 𝑥2 cos 𝑥 𝑑𝑥 (d) ∫ (𝑥2 + 𝑥 + 2) 𝑒2𝑥 𝑑𝑥

ln 𝑥
(e) ∫ ln 𝑥 𝑑𝑥 (f) ∫ 𝑑𝑥
𝑥2
1
𝑒𝑥 1
(g) ∫ 3 𝑑𝑥 (h) ∫ 𝑑𝑥
𝑥 𝑥2 − 7
1 𝑥3
(i) ∫ 𝑑𝑥 (j) ∫ 𝑑𝑥
5 − 4𝑥2 𝑥+3
𝑥−3 𝑥+2
(k) ∫ 3 𝑑𝑥 (l) ∫ 3 𝑑𝑥
𝑥 −𝑥 𝑥 − 2𝑥2
𝑥4 + 2 1
(m) ∫ 𝑑𝑥 (n) ∫ 𝑑𝑥
𝑥2 + 1 𝑥2 − 6𝑥 + 13

4. Find the following definite integrals.

1 2
2
(a) ∫ (1 − 2𝑥 − 3𝑥2 ) 𝑑𝑥 (b) ∫ (𝑥3 − 1) 𝑑𝑥
0 0
1 1
(c) ∫ (𝑥 − 1)(3𝑥 + 2) 𝑑𝑥 (d) ∫ 𝑥(𝑥2 + 1)5 𝑑𝑥
−1 0
1 1
(e) ∫ 𝑥𝑒𝑥 (f) ∫ 𝑥2 cos 𝑥3 𝑑𝑥
2
+1
𝑑𝑥
0 0
2 0
1
(g) ∫ 𝑑𝑥 (h) ∫ 𝑒𝑥+1 𝑑𝑥
2𝑥 + 3
0 −1

22
 𝜋
2 2
2
ln 2𝑥
(i) ∫ sin 𝑥(cos 2𝑥) 𝑑𝑥 (j) ∫ 𝑑𝑥
0 1
𝑥
𝑒𝜋 𝑒2
sin(ln 𝑥) 1
(k) ∫ 𝑑𝑥 (l) ∫ 𝑑𝑥
𝑥 𝑥 ln 𝑥
1 𝑒
1 1
(m) ∫ (𝑥 − 1)(𝑥2 − 2𝑥)4 𝑑𝑥 (n) ∫ 𝑥(1 − 𝑥)7 𝑑𝑥
−1 0
8 3
𝑥
(o) ∫ √ 𝑑𝑥 (p) ∫ √𝑥5 + 2 𝑑𝑥
𝑥+1
0 3
−𝑒 3
3
(q) ∫ 𝑑𝑥 (r) ∫ |𝑥2 − 1| 𝑑𝑥
−𝑒2
𝑥 −2
2 2
(s) ∫ (𝑥 − 2|𝑥|) 𝑑𝑥 (t) ∫ (𝑥2 − |𝑥 − 1|) 𝑑𝑥
−1 0

23