Definition of the derivative Knowledge-Based Download PDF

Questions

1. 1.

   Let the function $f$ be defined by $f(x) = \begin{cases} x^2 \cos\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$. What is the value of $f'(0)$?

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

   Let $f$ be a function defined for all real numbers such that $f(x + y) = f(x) + f(y) + 3xy$ for all real numbers $x$ and $y$. If $\lim_{h \to 0} \frac{f(h)}{h} = 5$, what is the value of $f'(2)$?

   • A) 11
   • B) 6
   • C) 16
   • D) 5
3. 3.

   Let $f$ be a differentiable function such that $f'(2) = 6$. What is the value of $\lim_{h \to 0} \frac{f(2 + 3h) - f(2 - h)}{h}$?

   • A) 24
   • B) 18
   • C) 12
   • D) 6
4. 4.

   Given that a function $f$ is differentiable at $x = 5$ with $f(5) = 3$ and $f'(5) = -4$, what is the value of the limit $\lim_{x \to 5} \frac{x f(5) - 5 f(x)}{x - 5}$?

   • A) -23
   • B) -17
   • C) 7
   • D) 23
5. 5.

   The limit $\lim_{h \to 0} \frac{\cos\left(\frac{\pi}{3} + h\right) - \frac{1}{2}}{h}$ represents the derivative of a function $f(x)$ at $x = a$. What is the value of this limit?

   • A) $\frac{\sqrt{3}}{2}$
   • B) $-\frac{1}{2}$
   • C) $\frac{1}{2}$
   • D) $-\frac{\sqrt{3}}{2}$