# 231 Review Problems

A solid foundation of understanding from MATH 231 will be key to your success in MATH 232. To help you review and strengthen your MATH 231 skills, your first homework assignment is to complete the Chapter Review, Limit, and Derivative problems below, in preparation for your first Mastery Quiz.

None of this will be collected, so you only have to do the parts that you need to complete in order to achieve mastery of these problems before the Mastery Quiz.

### Chapter Review Exercises

Compete the following sections of Chapter Review sets from the textbook.

• Chapter 0: Functions and Precalculus
• Notation and Algebraic Rules (p.90)
• Basic Algebra and Simple Functions (p.91)
• Chapter 1: Limits
• Limit Rules and Indeterminate Forms (p.161)
• Basic Limits (p.162)
• Chapter 2: Derivatives
• Notation and Differentiation Rules (p.221)
• Basic Derivatives (p.221)
• Chapter 3: Applications of the Derivative
• Geometric Formulas and Theorems (p.285)
• Derivatives and Curve Sketching (p.286)

### Limits Practice

Calculate each of the following limits algebraically.

1. $$\;\displaystyle \lim_{x \to 0^+} 2 x^{\frac{-3}{4}}$$
2. $$\;\displaystyle \lim_{x \to 2} \frac{4+2x}{x+2}$$
3. $$\;\displaystyle \lim_{x \to 1} \frac{1}{x^2-1}$$
4. $$\;\displaystyle \lim_{x \to -\infty} (5-2x+3x^3)$$
5. $$\;\displaystyle \lim_{x \to \infty} 4x^{-3}$$
6. $$\;\displaystyle \lim_{x \to \infty} \frac{\sqrt{x}}{1-\sqrt{x}}$$
7. $$\;\displaystyle \lim_{x \to \infty} \frac{(3x+1)^2(x-1)}{(1-x^3)}$$
8. $$\;\displaystyle \lim_{x \to 0} (3x^{-1}-2x^{-2})$$
9. $$\;\displaystyle \lim_{x \to 2} \frac{x+1}{(x-2)^2}$$
10. $$\;\displaystyle \lim_{x \to 1} \frac{x-1}{x^2-2x+1}$$
11. $$\;\displaystyle \lim_{x \to \infty} (\sqrt{x}-x)$$
12. $$\;\displaystyle \lim_{x \to 1} \frac{x+x^2-2x^3}{x-x^2}$$
13. $$\;\displaystyle \lim_{h \to 0} \frac{(2+h)^3-2^3}{h}$$
14. $$\;\displaystyle \lim_{h \to 0} \frac{\frac{1}{2+h}-\frac{1}{2}}{h}$$
15. $$\;\displaystyle \lim_{x \to \infty} (-2x^3+x^2-10)$$
16. $$\;\displaystyle \lim_{x \to \infty} \frac{x^{-3}}{x^2-x^{-1}}$$

### Derivatives Practice

Calculate the derivatives of each of the following functions.

1. $$\;\displaystyle f(x)=\sqrt{(3x^4-1)^{3}}$$
2. $$\;\displaystyle f(x)=\sqrt{(3x^4-1)^{3}+x}$$
3. $$\;\displaystyle f(x)=\frac{\sqrt{1-x}}{x^2-4}$$
4. $$\;\displaystyle f(x)=\sqrt{x}(5x+2)^{100}$$
5. $$\;\displaystyle f(x)=\sqrt{x(5x+2)^{100}}$$
6. $$\;\displaystyle f(x)=(\sqrt{x}(5x+2))^{100}$$
7. $$\;\displaystyle f(x)=\frac{x^5 + x\sqrt{x}}{x^2}$$
8. $$\;\displaystyle f(x)=\frac{1}{\sqrt{x}} + \frac{1}{x^2}$$
9. $$\;\displaystyle f(x)=\sqrt{\sqrt{x}}$$
10. $$\;\displaystyle f(x)=\frac{3}{x^{-\frac{3}{2}}\sqrt{x}}$$
11. $$\;\displaystyle f(x)=(3x+1)^2(2x+3)^8(5x-2)^4$$
12. $$\;\displaystyle f(x)=\frac{(x-1)(x-2)}{(x-3)(x-4)}$$
13. $$\;\displaystyle f(x)=(x^2-17x)^{-9} \cdot \frac{x^2+1}{\sqrt{2x+1}}$$
14. $$\;\displaystyle f(x)=(((x^2+1)^2+1)^2+1)^2$$
15. $$\;\displaystyle 3x^2+4y^2+xy=0 \;\;\;$$ (find $$\frac{dy}{dx}$$)
16. $$\;\displaystyle \frac{y^3+1}{x^3+1} = y^2 \;\;\;$$ (find $$\frac{dy}{dx}$$)
17. $$\;\displaystyle \frac{1}{y}-\frac{1}{x}=\frac{x^3}{y-1}\;\;\;$$ (find $$\frac{dy}{dx}$$)
18. $$\;\displaystyle A(t) = \pi (r(t))^2 \;\;\;$$ (find $$\frac{dA}{dr}$$ and $$\frac{dA}{dt}$$)
19. $$\;\displaystyle f(x)=\frac{1}{x^2+1}\;\;\;$$ (find $$f'(x), f”(x),$$ and $$f”'(x)$$)
20. $$\;\displaystyle f(x)=10x^8+6x^5-4x^2+17\;\;\;$$ (find $$f'(x)$$ and $$f^{[8]}$$ and $$f^{[9]}(x)$$)