AP Calculus Review #4

For a particle moving along the x-axis, $x\left(3\right)=-10$  and  $v\left(3\right)=-9$  .  At time t = 3, it can be said that the particle is ...

• Slowing down

• Moving away from the origin

• Moving toward the origin

• Speeding up

Answer explanation

When the position and velocity have the same sign, the particle is moving away from the origin.

The graph of f has a critical point when f'...

• is undefined or equals zero

• changes signs

• equals zero (only)

• is undefined (only)

Evaluate  $\lim_{h\rightarrow0}\ \frac{\cos\left(\frac{7\pi}{6}+h\right)-\cos\left(\frac{7\pi}{6}\right)}{h}$  

•  $\frac{\sqrt{3}}{2}$  

•  $\frac{1}{2}$  

•  $-\frac{\sqrt{3}}{2}$  

•  $-\frac{1}{2}$  

Answer explanation

This is saying to find the derivative of cosx evaluated at  $x=\frac{7\pi}{6}$  

The graph of f has a relative extrema when f' ...

• Changes signs

• goes from + to -

• goes from - to +

• equals zero

Answer explanation

A relative extrema is a max or min

If  $g\left(x\right)=f^{-1}\left(x\right)$  , with  $f\left(6\right)=2\ \&\ g\left(3\right)=2$  ,

then  $g'\left(2\right)=?$  

•  $f'\left(3\right)$  

•  $\frac{1}{f'\left(6\right)}$  

•  $\frac{1}{f'\left(3\right)}$  

•  $f'\left(6\right)$  

Answer explanation

 $g'\left(x\right)=\frac{1}{f'\left(g\left(x\right)\right)}$  

Let  $g\left(x\right)=\int_0^x\ f\left(t\right)dt$  , where the graph of f is shown below.  On what interval(s) is g both concave down and increasing?

• (-5, -2)

• (5, 8)

• cannot be determined

• (-5, -2) U (0, 2)

Answer explanation

Concave down:  the graph of f''<0 OR f' slopes are negative
Increasing:  the graph of f' is above the x-axis

 $\int\sin x\ dx$  =

•  $\frac{1}{\sqrt{1-x^2}}+c$  

•  $-\cos x\ +c$  

•  $\cos x\ +c$  

•  $-\frac{1}{\sqrt{1-x^2}}+c$  

Answer explanation

Integral means ANTI derivative

If  $g\left(x\right)=f^{-1}\left(x\right)$  , with  $f\left(6\right)=5$  and  $g\left(6\right)=8$  , then  $g'\left(6\right)=?$  

•  $\frac{1}{f'\left(5\right)}$  

•  $\frac{1}{g'\left(8\right)}$  

•  $\frac{1}{f'\left(8\right)}$  

•  $\frac{1}{g'\left(5\right)}$  

Answer explanation

 $g'\left(x\right)=\frac{1}{f'\left(g\left(x\right)\right)}$  

Let  $g\left(x\right)=\int_0^x\ f\left(t\right)dt$  , where the graph of f is shown.  For what value(s) of x does g have a relative minimum?

• -2 and 2

• 8

• -2 and 5

• 2

Answer explanation

Max and Mins are where the graph of the derivative crosses the x-axis either +/- or -/+.  In this case -/+ since wer are looking for a minimum.

For a particle moving along the x-axis,  $x\left(3\right)=-6$  and  $v\left(3\right)=-7$  .  At time t = 3, it can be said that the particle is...

• Moving toward the origin

• Slowing down

• Moving away from the origin

• Speeding up

Answer explanation

When the position and velocity have the same signs, the particle is moving AWAY from the origin.