Definition of the derivative Knowledge-Based Practice

Questions

1. 1.

   Let $f$ be a function differentiable at $x = 1$ such that $f'(1) = 5$. What is the value of the limit $\lim_{h \to 0} \frac{f(1+2h) - f(1-h)}{h}$?

   • A) 5
   • B) 10
   • C) 15
   • D) 20
2. 2.

   What is the value of the limit $\lim_{h \to 0} \frac{\frac{1}{2+h} - \frac{1}{2}}{h}$?

   • A) $\frac{1}{4}$
   • B) $-\frac{1}{2}$
   • C) $-\frac{1}{4}$
   • D) $0$
3. 3.

   What is the value of the limit $\lim_{h \to 0} \frac{\sqrt{9+h} - 3}{h}$?

   • A) $\frac{1}{3}$
   • B) $\frac{1}{6}$
   • C) $\frac{1}{12}$
   • D) $6$
4. 4.

   What is the value of the limit $\lim_{x \to \pi} \frac{\sin(x)}{x - \pi}$?

   • A) 1
   • B) 0
   • C) Undefined
   • D) $-1$
5. 5.

   What is the value of the limit $\lim_{h \to 0} \frac{(2+h)^4 - 16}{h}$?

   • A) 8
   • B) 16
   • C) 32
   • D) 64

Answers and step-by-step solutions appear after you finish the test.