Calculus review 1

 $\lim\ \frac{4x^2\ }{x^2\ +\ 10000x}$  

 $x\rightarrow\infty$  

• 0

•  $\frac{1}{2500}$  

• 1

• 4

• nonexistant

If the function f is continuous for all real numbers and if f(x) =  $\frac{x^2-4}{x+2}$   when x  $\ne$  -2 then f(-2) =   
   

• -4

• -2

• -1

• 0

• 2

If  as x  $\rightarrow$  a,    the lim f(x) = L, where L is a real number, which of the following must be true?

• f'(a) exists

• f(x) is continuous at x = a

• f(x) is defined at x =a 
  
  

• f(a) = L

• None of the above

If f(x) = x, then f'(5) =

• 0

• $\frac{1}{5}$

• 1

• 5

• 25

 $If\ x^2+xy+y^{3\ }=0\ then,\ in\ terms\ of\ x\ and\ y,\ \frac{dy}{dx}=$  

•  $-\frac{2x\ +\ y}{x\ +\ 3y^2}$  

•  $-\frac{x+3y^2}{2x+y}$  

•  $\frac{-2x}{1\ +3y^2}$  

•  $\frac{-2x}{x\ +3y^2}$  

•  $-\frac{2x\ +y}{x+3y^2-1}$  

 $\frac{d}{dx}\left(\frac{1}{x^3}-\frac{1}{x}+x^2\right)\ at\ x\ =\ -1\ is$  

• -6

• -4

• 0

• 2

• 6

 $If\ f\left(x\right)\ =\ \sin x\ then\ f'\left(\frac{\pi}{3}\right)\ =$  

•  $\frac{-1}{2}$  

•  $\frac{1}{2}$  

•  $\frac{\sqrt{2}}{2}$  

• 1

•  $\sqrt{2}$  

 $If\ f\left(x\right)\ =\ \sqrt{2x\ }\ ,\ then\ f'\left(2\right)$  

•  $\frac{1}{4}$  

•  $\frac{1}{2}$  

•  $\frac{\sqrt{2}}{2}$  

• 1

•  $\sqrt{2}$  

 $If\ f\left(x\right)\ =\ \left(x-1\right)^2\ \sin x,\ then\ f'\left(0\right)=$  

• -2

• -1

• 0

• 1

• 2

 $e^{f\left(x\right)}\ =\ 1\ +x^2,\ then\ f'\left(x\right)\ =$  

•  $\frac{1}{1\ +\ x^2}$  

•  $\frac{2x}{1\ +\ x^2}$  

•  $2x\left(1+x^2\right)$  

•  $2x\left(e^{\left(1+x^2\right)}\right)$  

•  $2x\ln\left(1+x^2\right)$