5.12 Continuity, differentiability and first principles Quiz

1. For $f(x) = 3x^{2}$, which expression equals $\dfrac{f(x+h) - f(x)}{h}$ for $h \neq 0$?

• A. $6x$
• B. $6x + 3h$
• C. $6x + h$
• D. $6xh + 3h^{2}$

2. Which function is continuous at $x = 1$ but not differentiable there?

• A. $y = (x-1)^{2}$
• B. $y = \dfrac{1}{x - 1}$
• C. $y = |x - 1|$
• D. $y = x^{3}$

3. Find $\lim_{x \to -1}\dfrac{x^{2} + 3x + 2}{x + 1}$.

• A. $0$
• B. $1$
• C. $2$
• D. It does not exist

4. If $y = x^{4} - 2x^{3}$, find $\dfrac{d^{3}y}{dx^{3}}$.

• A. $24x - 12$
• B. $12x^{2} - 12x$
• C. $24$
• D. $4x^{3} - 6x^{2}$

5. Which limit diverges?

• A. $\lim_{x \to \infty}\dfrac{1}{x}$
• B. $\lim_{x \to 0}\dfrac{x^{2} + x}{x}$
• C. $\lim_{x \to 2}(x^{2} + 1)$
• D. $\lim_{x \to 0}\dfrac{1}{x^{2}}$