AP Calculus Review

The graph of f has a relative extrema when f'

• changes sign

• equals zero

• goes from negative to positive

• goes from positive to negative

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

Let $g\left(x\right)=\int_0^xf\left(t\right)dt$ , where the graph of f is shown above.  On what interval(s) is g decreasing?

• [2,8]

• [0,5]

• [2,5]

• [-5,-2]U[0,5]

Answer explanation

since g'=f, g is decreasing when f is negative.

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

If $w\left(x\right)=f\left(x\right)q\left(x\right)$ , then  $w'\left(x\right)=$   

•  $f'\left(x\right)q'\left(x\right)$  

•  $f'\left(x\right)q'\left(x\right)+f\left(x\right)q\left(x\right)$  

•  $f'\left(q\left(x\right)\right)q'\left(x\right)$  

•  $f'\left(x\right)q\left(x\right)+f\left(x\right)q'\left(x\right)$  

Answer explanation

product rule!

For the function f, it is known that $\lim_{x\rightarrow-1^-}f'\left(x\right)=\lim_{x\rightarrow-1^+}f'\left(x\right)$  .  Which of the following must be true?

I.  f is continuous at x = -1

II.  f is differentiable at x = -1

• Both I and II

• I only

• II only

• Neither I or II

Answer explanation

Since we know the right and left hand limits are equal, the limit must exist, but since we do not know whether this limit equals f(-1), we cannot conclude with certainty that f is continuous at x=-1.  Since continuity cannot be established, we cannot establish differentiability, which has continuity as a requirement.

Evaluate $\lim_{x\rightarrow\frac{\pi}{4}}\frac{\cos x-\cos\left(\frac{\pi}{4}\right)}{x-\frac{\pi}{4}}$  

•  $-\frac{\sqrt{2}}{2}$  

• DNE

•  $0$  

•  $\frac{\sqrt{2}}{2}$  

Tags

CCSS.HSF.TF.A.3

 $\frac{d}{dx}\left(\frac{1}{1-2x}\right)$  equals

•  $-\frac{2}{\left(1-2x\right)^2}$  

•  $-2\ln\left|1-2x\right|$  

•  $\frac{2}{\left(1-2x\right)^2}$  

•  $-\frac{1}{2}\ln\left|1-2x\right|$  

Answer explanation

Taking the derivative will not use ln. Use the power and chain rule:

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.IF.C.8

For a particle moving along the x-axis, the particle's speed equals 2 when

• $\left|v\left(t\right)\right|=2$

• $v\left(t\right)=2$

• $a\left(t\right)=2$

• $\left|a\left(t\right)\right|=2$

Answer explanation

Speed is the absolute value of velocity

Tags

CCSS.HSA.SSE.A.1

CCSS.HSN.VM.A.3

If $f\left(x\right)=\int_4^x\left(\cos\left(2t\right)+3\right)dt$  , then  $f'\left(x\right)=$  

•  $\frac{1}{2}\sin\left(2x\right)+3x$  

•  $\cos\left(2x\right)$  

•  $2\cos\left(2x\right)+3$  

•  $\cos\left(2x\right)+3$  

 $\frac{d}{dx}\cos x=$  

•  $-\frac{1}{\sqrt{1-x^2}}$  

•  $\frac{1}{\sqrt{1-x^2}}$  

•  $-\sin x$  

•  $\sin x$  

The region R is the area enclosed by the functions $y=2x$  and  $y=x^2$  .  Which expression represents the area of R?

•  $\int_0^2\left(x^2-2x\right)dx$  

•  $\int_0^2\left(2x-x^2\right)dx$  

•  $\int_0^4\left(2x-x^2\right)dx$  

•  $\int_0^4\left(x^2-2x\right)dx$  

Tags

CCSS.HSA.REI.C.7

CCSS.HSA.REI.D.11