Gradient of a curve and differentiation from first principles Quiz

1. Find the gradient of the chord joining the points on $y=x^2$ with $x=2$ and $x=2.1$.

• A. $4$
• B. $0.41$
• C. $4.1$
• D. $4.41$

2. From first principles, $\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}=4x+2h$ for $\mathrm{f}(x)=2x^2$. What is $\mathrm{f}'(x)$?

• A. $4x$
• B. $4x+2h$
• C. $2x$
• D. $0$

3. If $y=x^3$, what is $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$?

• A. $3x^2$
• B. $9x^4$
• C. $3x$
• D. $6x$

4. A curve has a minimum point at $x=1$. What is true of $\mathrm{f}'(x)$?

• A. $\mathrm{f}'(x)>0$ for all $x$
• B. $\mathrm{f}'(1)=0$, negative for $x<1$ and positive for $x>1$ near 1
• C. $\mathrm{f}'(1)$ is the minimum value of $\mathrm{f}'$
• D. $\mathrm{f}'(x)<0$ for $x>1$

5. The volume of a block of ice is modelled by $V=t^3$ cm³ at time $t$ minutes. Using the result for $x^3$, find the rate of change of $V$ at $t=2$.

• A. $6$
• B. $8$
• C. $12$
• D. $3$