Differentiation Exam

Find an equation of the tangent line to the graph of  $x^2+2y^2=3$  at the point  $\left(1,-1\right)$  

•  $y+1=\frac{1}{2}\left(x-1\right)$  

•  $y+1=-2\left(x-1\right)$  

•  $y+1=-\frac{1}{2}\left(x-1\right)$  

•  $y+1=2\left(x-1\right)$  

If $e^{xy}-y^2=e-4$ , then at  $x=\frac{1}{2}$  and  $y=2,\frac{dy}{dx}=$  

•  $\frac{e}{2}$  

•  $\frac{e}{4}$  

•  $\frac{4e}{4-e}$  

•  $\frac{4e}{8-e}$  

Suppose that  $g\left(x\right)=h\left(f\left(x\right)\right)$ ,  $f\left(x\right)=6$ ,  $h\left(3\right)=2$ ,  $h'\left(6\right)=4$ ,  $h'\left(3\right)=15$ , and  $f'\left(3\right)=8$ . Find  $g'\left(3\right)$ .

•  $60$  

•  $32$  

•  $4$  

•  $120$  

Suppose that  $g\left(x\right)=\frac{h\left(x\right)}{f\left(x\right)}$ ,  $f\left(x\right)=6$ ,  $h\left(3\right)=2$ ,  $h'\left(6\right)=4$ ,  $h'\left(3\right)=15$ , and  $f'\left(3\right)=8$ . Find  $g'\left(3\right)$ .

•  $\frac{1}{3}$  

•  $3$  

•  $2$  

•  $\frac{5}{3}$  

Suppose that  $g\left(x\right)=h\left(x\right)\cdot f\left(x\right)$ ,  $f\left(x\right)=6$ ,  $h\left(3\right)=2$ ,  $h'\left(6\right)=4$ ,  $h'\left(3\right)=15$ , and  $f'\left(3\right)=8$ . Find  $g'\left(3\right)$ .

•  $32$  

•  $8$  

•  $106$  

•  $60$  

At  $x=2$ , the function shown is:

• both continuous and  differentiable

• continuous, but not differentiable

• differentiable, but not continuous

• neither continuous or differentiable

A counter records people entering a fast-food restaurant. The data below shows the number of customers that enter the restaurant for each two-hour period starting at 8:00AM. Estimate the rate at which the customers are entering the store at 1:00PM. ( $t=5$ )

•  $6\ \frac{customers}{hour}$  

•  $-3\ \frac{customers}{hour}$  
  

•  $39\ \frac{customers}{hour}$ 

•  $-6\ \frac{customers}{hour}$  
  

Given $h\left(x\right)=f\left(g\left(x\right)\right)$ , use the graph to find  $h'\left(4\right)$ .

•  $\frac{5}{8}$  

•  $\frac{1}{2}$  

•  $5$  

•  $\frac{5}{4}$  

If  $f\left(x\right)=\ln\left(x-\frac{1}{x}\right)$ , then  $f'\left(x\right)=$  

•  $\frac{x^2+1}{x^3-x}$  

•  $\frac{x^2}{1-x^2}$  

•  $\frac{x^2+1}{x^2-1}$  

•  $\frac{x^2}{1+x^2}$  

Find  $f'\left(x\right)$  if   $f\left(x\right)=\sin^53x$ .

•  $f'\left(x\right)=5\sin^43x\cdot\cos3x$  

•  $f'\left(x\right)=60\sin^43x\cdot\cos3x$  

•  $f'\left(x\right)=15\cos^43x$  

•  $f'\left(x\right)=15\sin^43x\cdot\cos3x$