Derivative Rules and Graphs

f(x) will have a maximum when f'(x) changes from

• increasing to decreasing

• decreasing to increasing

• negative to positive

• positive to negative

Which graph represents the derivative of this graph?

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Tags

CCSS.HSF.BF.B.3

The graph of f'(x) is shown above.

f(x) is increasing over what interval?

• $\left(-\infty,-5\right)\cup\left(-2,1\right)$

• $\left(-5,-2\right)\cup\left(1,\infty\right)$

• $\left(-\infty,-4\right)\cup\left(0,\infty\right)$

• $\left(-4,0\right)$

If f'(x) is increasing, then f''(x) is

• positive

• negative

• increasing

• decreasing

What is the derivative of  $\frac{f\left(x\right)}{g\left(x\right)}$  ?

•  $\frac{g'\left(x\right)f\left(x\right)+f'\left(x\right)g\left(x\right)}{\left(g\left(x\right)\right)^2}$  

•  $\frac{g\left(x\right)f'\left(x\right)+f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^2}$  

•  $\frac{g'\left(x\right)f\left(x\right)-f'\left(x\right)g\left(x\right)}{\left(g\left(x\right)\right)^2}$  

•  $\frac{g\left(x\right)f'\left(x\right)-f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^2}$  

The graph of f is shown above.

Which graph represents the graph of f'(x)?

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Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

 $\lim_{h\rightarrow0}\ \frac{2\left(3+h\right)^2-4\left(3+h\right)-6}{h}=$  

• -6

• 6

• 8

• 10

What is the equation of the line tangent to the graph of $f\left(x\right)=5x^3-3x^2-10$  at  $x=2$  ?

•  $y-18=48\left(x-2\right)$  

•  $y-48=18\left(x-2\right)$  

•  $y-4=48\left(x-2\right)$  

•  $y-4=18\left(x-2\right)$  

 If  $f\left(x\right)$  is continuous and $f'\left(x\right)<0$ to the left of  $x=5$  and  $f'\left(x\right)>0$  to the right of  $x=5$ , then  $f\left(x\right)$  has a _________ at  $x=5$ .

• maximum

• minimum

• zero

• point of tangency

What is the average rate of change of the function $f\left(x\right)=x^3-2x^2-x+1$  over the interval  $\left[1,3\right]$ ?

•  $\frac{1}{4}$  

•  $-\frac{1}{4}$  

• 4

• -4