Differentiation from first principles Quiz

1. For $f(x)=x^{2}$, simplify $\frac{f(x+h)-f(x)}{h}$.

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

2. Which statement correctly describes $\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}$?

• A. The rate of change of the gradient of the curve
• B. The gradient of the tangent squared
• C. The area under the curve
• D. The value of $y$ squared

3. Which is the first-principles definition of $f'(x)$?

• A. $\lim_{h\to0}\frac{f(x)+h}{h}$
• B. $\lim_{h\to0}f(x+h)-f(x)$
• C. $\frac{f(x+h)-f(x)}{h}$ with $h=0$
• D. $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$

4. Using $f'(x)=3x^{2}$ for $f(x)=x^{3}$, what is the gradient of the curve at $x=2$?

• A. $6$
• B. $12$
• C. $8$
• D. $4$

5. A curve has a minimum at $x=2$. What is true of $f'(x)$ near $x=2$?

• A. It is positive on both sides of $x=2$
• B. It is negative on both sides of $x=2$
• C. It is zero at $x=2$, changing from negative to positive
• D. It is zero at $x=2$, changing from positive to negative