Revision: Differentiation & Integration

Find  $\frac{\text{d}x}{\text{d}t}$  when the radius of the rod is 2 cm, giving your answer to 3 significant figures. 

The point A, on the curve C   $y=x^{4\ }-8x^{2\ }+3$  , has x-coordinate 1. Find an equation for the normal to C at A, giving your answer in the form ax+by+c=0, where a, b and c are integers.

The figure shows part of the curve C with equation.   $y=3x^{\ \frac{1}{2\ }}-x^{\frac{3}{2}},\ \ \ x\ge0$  . The point A on C is a stationary point and C cuts the x-axis at the point B. Given x-coordinate of B is 3, find the exact area of the finite region closed by C and the x-axis, shown shaded in the figure. Give your answer correct to 3 significant figures.

Find $\int_{ }^{ }2x^2-\frac{6}{x^3\ }\ dx,\ \ x\ne0.$  

•  $y=\frac{2}{3}x^3-\frac{3}{2x^2}+C$  

•  $y=\frac{x^2}{3}-\frac{3}{x^4}+C$  

•  $y=2x^{3\ }-\frac{6}{x^4}+C$  

•  $y=\frac{2}{3}x^{3\ }+\frac{3}{x^{2\ }}+C$  

Find $\int_0^3\ x^{3\ }-10x^2+28x\ dx$  . Leave your answer in exact values or in 3 significant figures. 

Differentiate $y=\left(x-1\right)\left(x^2-4\right)$ .

•  $\frac{dy}{dx}=x^{3\ }-x^2-4x-4$ 

•  $\frac{dy}{dx}=3x^2-2x+4$ 

•   $\frac{dy}{dx}=3x^2-2x-4$

•  $\frac{dy}{dx}=2x^2-4x\ +3$ 

 $\frac{\text{d}}{\text{d}x}\left(\frac{3x^3-x^2}{x}\right)$  

•  $9x^2-2x$  

•  $6x-1$  

•  $6x$  

•  $9x^2-2$  

Find the second derivative of:

f(x) = 2x² - 8x + 9

• -8

• 4

• 9

• 0

Find the turning points of:

f(x)=x³+3x²-9x+1 (hint dy/dx=0)

• (-3,28) and (1,-4)

• (3,28) and (1,4)

• (1,28) and (-3,-4)

• (3,28) and (-1,-4)

 $\int\left(\frac{1}{x^2}+\frac{6}{x^3}\right)dx$ 

•  $-x^{-1}-3x^{-2}+C$ 

•  $\frac{x^{-3}}{-3}+\frac{6x^{-4}}{-4}+C$ 

•  $-x^{-1}-3x^{-2}$ 

•  $-x-3x^2+C$