## 1,001 Calculus Practice Problems

__Part I__

## The Questions

*In this part …*

The only way to become proficient in math is through a lot of practice. Fortunately, you have now 1,001 practice opportunities right in front of you. These questions cover a variety of calculus-related concepts and range in difficulty from easy to hard. Master these problems, and you'll be well on your way to a very solid calculus foundation.

Here are the types of problems that you can expect to see:

· Algebra review (Chapter __1__)

· Trigonometry review (Chapter __2__)

· Limits and continuity (Chapter __3__)

· Derivative fundamentals (Chapters __4__ through __7__)

· Applications of derivatives (Chapter __8__)

· Antiderivative basics (Chapters __9__ and __10__)

· Applications of antiderivatives (Chapter __11__)

· Antiderivatives of other common functions and L'Hôpital's rule (Chapter __12__)

· More integration techniques (Chapters __13__ and __14__)

· Improper integrals, the trapezoid rule, and Simpson's rule (Chapter __15__)

__Chapter 1__

### Algebra Review

Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps of algebraic simplification. Having a strong algebra background will allow you to focus on the calculus concepts and not get lost in the mechanical manipulation that's required to solve the problem.

*The Problems You'll Work On*

In this chapter, you see a variety of algebra problems:

· Simplifying exponents and radicals

· Finding the inverse of a function

· Understanding and transforming graphs of common functions

· Finding the domain and range of a function using a graph

· Combining and simplifying polynomial expressions

*What to Watch Out For*

Don't let common mistakes trip you up. Some of the following suggestions may be helpful:

· Be careful when using properties of exponents. For example, when multiplying like bases, you add the exponents, and when dividing like bases, you subtract the exponents.

· Factor thoroughly in order to simplify expressions.

· Check your solutions for equations and inequalities if you're unsure of your answer. Some solutions may be extraneous!

· It's easy to forget some algebra techniques, so don't worry if you don't remember everything! Review, review, review.

*Simplifying Fractions*

*1–13** Simplify the given fractions by adding, subtracting, multiplying, and/or dividing.*

**1.**

**2.**

**3.**

**4.**

**5.**

**6.**

**7.**

**8.**

**9.**

**10.**

**11.**

**12.**

**13.**

*Simplifying Radicals*

*14–18** Simplify the given radicals. Assume all variables are positive.*

**14.**

**15.**

**16.**

**17.**

**18.**

*Writing Exponents Using Radical Notation*

*19–20** Convert between exponential and radical notation.*

**19.** Convert to radical notation. (** Note:** The final answer can have more than one radical sign.)

**20.** Convert to exponential notation.

*The Horizontal Line Test*

*21–23** Use the horizontal line test to identify one-to-one functions.*

**21.** Use the horizontal line test to determine which of the following functions is a one-to-one function and therefore has an inverse.

(A) *y* = *x*^{2} + 4*x* + 6

(B)

(C)

(D) *y* = 3*x* + 8

(E)

**22.** Use the horizontal line test to determine which of the following functions is a one-to-one function and therefore has an inverse.

(A) *y* = *x*^{2} – 4

(B) *y* = *x*^{2} – 4, *x* ≥ 0

(C) *y* = *x*^{2} – 4, –2 ≤ *x* ≤ 8

(D) *y* = *x*^{2} – 4, –12 ≤ *x* ≤ 6

(E) *y* = *x*^{2} – 4, –5.3 ≤ *x* ≤ 0.1

**23.** Use the horizontal line test to determine which of the following functions is a one-to-one function and therefore has an inverse.

(A) *y* = *x*^{4} + 3*x*^{2} – 7

(B)

(C) *y* = cos *x*

(D) *y* = sin *x*

(E) *y* = tan^{−1 }*x*

*Find Inverses Algebraically*

*24–29** Find the inverse of the one-to-one function algebraically.*

**24.** *f* (*x*) = 4 – 5*x*

**25.** *f* (*x*) = *x*^{2} – 4*x*, *x* ≥ 2

**26.**

**27.** *f* (*x*) = 3*x*^{5} + 7

**28.**

**29.**

*The Domain and Range of a Function and Its Inverse*

*30–32** Solve the given question related to a function and its inverse.*

**30.** The set of points {(0, 1), (3, 4), (5, –6)} is on the graph of *f* (*x*), which is a one-to-one function. Which points belong to the graph of *f*^{ −1}(*x*)?

**31.** *f* (*x*) is a one-to-one function with domain [–2, 4) and range (–1, 2). What are the domain and range of *f*^{ −1}(*x*)?

**32.** Suppose that *f* (*x*) is a one-to-one function. What is an expression for the inverse of *g*(*x*) = *f* (*x* + *c*)?

*Linear Equations*

*33–37** Solve the given linear equation.*

**33.** 3 *x* + 7 = 13

**34.** 2 (*x* + 1) = 3(*x* + 2)

**35.** –4(*x* + 1) – 2*x* = 7*x* + 3(*x* – 8)

**36.**

**37.**

*Quadratic Equations*

*38–43** Solve the quadratic equation.*

**38.** Solve *x*^{2} – 4*x* – 21 = 0.

**39.** Solve *x*^{2} + 8*x* – 17 = 0 by completing the square.

**40.** Solve 2*x*^{2} + 3*x* – 4 = 0 by completing the square.

**41.** Solve 6x^{2} + 5x – 4 = 0.

**42.** Solve 3*x*^{2} + 4x – 2 = 0.

**43.** Solve x^{10} + 7x^{5} + 10 = 0.

*Solving Polynomial Equations by Factoring*

*44–47** Solve the polynomial equation by factoring.*

**44.** 3 *x*^{4} + 2*x*^{3} – 5*x*^{2} = 0

**45.** *x*^{8} + 12*x*^{4} + 35 = 0

**46.** *x*^{4} + 3*x*^{2} – 4 = 0

**47.** *x*^{4} – 81 = 0

*Absolute Value Equations*

*48–51** Solve the given absolute value equation.*

**48.**

**49.**

**50.**

**51.**

*Solving Rational Equations*

*52–55** Solve the given rational equation.*

**52.**

**53.**

**54.**

**55.**

*Polynomial and Rational Inequalities*

*56–59** Solve the given polynomial or rational inequality.*

**56.** *x*^{2} – 4*x* – 32 < 0

**57.** 2 *x*^{4} + 2*x*^{3} ≥ 12*x*^{2}

**58.**

**59.**

*Absolute Value Inequalities*

*60–62** Solve the absolute value inequality.*

**60.**

**61.**

**62.**

*Graphing Common Functions*

*63–77** Solve the given question related to graphing common functions.*

**63.** What is the slope of the line that goes through the points (1, 2) and (5, 9)?

**64.** What is the equation of the line that has a slope of 4 and goes through the point (0, 5)?

**65.** What is the equation of the line that goes through the points (–2, 3) and (4, 8)?

**66.** Find the equation of the line that goes through the point (1, 5) and is parallel to the line .

**67.** Find the equation of the line that goes through the point (3, –4) and is perpendicular to the line that goes through the points (3, –4) and (–6, 2).

**68.** What is the equation of the graph of after you stretch it vertically by a factor of 2, shift the graph 3 units to the right, and then shift it 4 units upward?

**69.** Find the vertex form of the parabola that passes through the point (0, 2) and has a vertex at (–2, –4).

**70.** Find the vertex form of the parabola that passes through the point (1, 2) and has a vertex at (–1, 6).

**71.** A parabola has the vertex form *y* = 3(*x* + 1)^{2} + 4. What is the vertex form of this parabola if it's shifted 6 units to the right and 2 units down?

**72.** What is the equation of the graph of *y* = *e ^{x}* after you compress the graph horizontally by a factor of 2, reflect it across the

*y*-axis, and shift it down 5 units?

**73.** What is the equation of the graph of after you stretch the graph horizontally by a factor of 5, reflect it across the *x*-axis, and shift it up 3 units?

**74.** Find the equation of the third-degree polynomial that goes through the points (–4, 0), (–2, 0), (0, 3), and (1, 0).

**75.** Find the equation of the fourth-degree polynomial that goes through the point (1, 4) and has the roots –1, 2, and 3, where 3 is a repeated root.

**76.** A parabola crosses the *x*-axis at the points (–4, 0) and (6, 0). If the point (0, 8) is on the parabola, what is the equation of the parabola?

**77.** A parabola crosses the *x*-axis at the points (–8, 0) and (–2, 0), and the point (–4, –12) is on the parabola. What is the equation of the parabola?

*Domain and Range from a Graph*

*78–80** Find the domain and range of the function with the given graph.*

**78.**

**79.**

**80.**

*End Behavior of Polynomials*

*81–82** Find the end behavior of the given polynomial. That is, find and .*

**81.** f(x) = 3x^{6} – 40x^{5} + 33

**82.** f(x) = –7x^{9} + 33x^{8} – 51x^{7} + 19x^{4} – 1

*Adding Polynomials*

*83–87** Add the given polynomials.*

**83.** (5*x* + 6) + (–2*x* + 6)

**84.** (2*x*^{2} – *x* + 7) + (–2*x*^{2} + 4*x* – 9)

**85.** (*x*^{3} – 5*x*^{2} + 6) + (4*x*^{2} + 2*x* + 8)

**86.** (3*x* + *x*^{4} + 2) + (–3*x*^{4} + 6)

**87.** (*x*^{4} – 6*x*^{2} + 3) + (5*x*^{3} + 3*x*^{2} – 3)

*Subtracting Polynomials*

*88–92** Subtract the given polynomials.*

**88.** (5*x* – 3) – (2*x* + 4)

**89.** (*x*^{2} – 3*x* + 1) – (–5*x*^{2} + 2*x* – 4)

**90.** (8*x*^{3} + 5*x*^{2} – 3*x* + 2) – (4*x*^{3} + 5*x* – 12)

**91.** (*x* + 3) – (*x*^{2} + 3*x* – 4) – (–3*x*^{2} – 5*x* + 6)

**92.** (10*x*^{4} – 6*x*^{3} + *x*^{2} + 6) – (*x*^{3} + 10*x*^{2} + 8*x* – 4)

*Multiplying Polynomials*

*93–97** Multiply the given polynomials.*

**93.** 5*x*^{2}(*x* – 3)

**94.** (*x* + 4)(3*x* – 5)

**95.** (*x* – *y* + 6)(*xy*)

**96.** (2*x* – 1)(*x*^{2} – *x* + 4)

**97.** –*x*(*x*^{4} + 3*x*^{2} + 2)(*x* + 3)

*Long Division of Polynomials*

*98–102** Use polynomial long division to divide.*

**98.**

**99.**

**100.**

**101.**

**102.**