MAF12 Revision for Exam

 Evaluate the limit for the function $\frac{x^2-4}{2x^2-x-6}$ as  $x\rightarrow2$  

• limit does not exist 

•  $\frac{4}{7}$  

•  $0$  

•  $\frac{7}{4}$  

Limit will only exist if the left side limit is not equal to the right side limit.

• True

• False

Find the derivative of the following function: $\frac{x}{\tan x}$  

•  $\frac{\tan x\ +x\sec^2x}{\tan^2x}$  

•  $\frac{\tan x\ -x\sec^2x}{\tan^2x}$  

•  $\frac{\tan x\ +x\tan^2x}{\sec^2x}$  

•  $\frac{\tan x\ -x\tan^2x}{\sec^2x}$  

A circular metal disk is heated and its radius is changing at a rate of  $0.02m^2$ /s. Find the rate at which the area will change when the radius is 150cm? The are of the disc is given by  $A=\pi r^2$  

\frac{0.02m^2}{s}

•  $\frac{dA}{dt}=2\pi r$  

•  $\frac{dA}{dt}=0.02$  

•  $\frac{dr}{dt}=0.018$  

•  $\frac{dr}{dt}=2\times10^{-5}$  

If $y=4\cos^5(3x−2)$  , then  $\frac{\text{d}x}{\text{d}y}$  is equal to

•  $-60\cos^4\left(3x-2\right)\sin\left(3x-2\right)$  

•  $-60\sin^4\left(3x-2\right)\cos\left(3x-2\right)$  
  

•  $60\cos^4\left(3x-2\right)\sin\left(3x-2\right)$  

•  $-20\cos^4\left(3x-2\right)\sin\left(3x-2\right)$  

Consider the parabola given by the equation 
 $y=x^2+4x-5$  . At which point on the graph of this parabola is the slope of the tangent line equal to 10?

Note: point refers to x and y value. therefore answer should be in the form  $\left(x,y\right)$ . 

•  $\left(1,0\right)$  

•  $\left(3,16\right)$  

•  $\left(2,7\right)$  

•  $\left(10,135\right)$  

Find the relative extreme of the function 

 $x^3-x^2-x+3$  

• No relative extreme 

• Max at  $\left(\frac{-1}{3},\frac{32}{27}\right)$  and Min at  $\left(1,2\right)$  

• Max at  $\left(1,2\right)$  and Min at  $\left(\frac{-1}{3},\frac{32}{27}\right)$  

• Max at  $\left(-\frac{1}{3},1\right)$  

The derivative of $y=\sqrt{\frac{x^2-1}{x^2+1}}$  is NOT Equal to 

 $\frac{\text{d}x}{\text{d}y}=\frac{2x}{\left(x^2+1\right)^{\frac{3}{2}}\left(x^2-1\right)^{\frac{1}{2}}}$  

• True 

• False 

State whether the given is true or false

The opposite of $\frac{\text{d}x}{\text{d}y}$  is not  $\int_{ }^{ }f\left(x\right)dx$  

• True

• False 

Find the area under the curve of the following function from $x=0\ to\ x=\pi$ .

 $f\left(x\right)=\cos^2\left(x\right)\sin\left(x\right)$   

• 0

• -1/3

• 1/3

• none of these

• 2/3