AP Calculus Review #2

 Selected values from the function f(x) are shown in the table. When a midpoint Riemann sum with two subintervals is used to complete $\int_2^{10}f\left(x\right)dx$ , the value is...

• 24

• 28

• 36

• 26

Answer explanation

Remember this involves adding the areas of rectangles.  A=base*height

 $\frac{d}{dx}\left(\frac{1}{2x+5}\right)=?$  

•  $-\frac{2}{\left(2x+5\right)^2}$  

•  $\frac{2}{\left(2x+5\right)^2}$  

•  $\frac{1}{2}\ln\left|2x+5\right|$  

•  $2\ln\left|2x+5\right|$  

Answer explanation

You have to use power rule with a negative exponent along with chain rule!

 $\frac{dy}{dx}=e^{x+3}y^3$  

What would be a step in solving the differential equation?

•  $\int y^3dy=\int e^{x+3}dx$  

•  $\int\ \frac{1}{y^3}dy=\int\ e^{x+3}dx$  

•  $\int\ \frac{1}{y^3}dy=\int\ \frac{1}{e^{x+3}}dx$  

•  $\int\ y^3dy=\int\ \frac{1}{e^{x+3}}dx$  

Answer explanation

Implicit differentiation

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

• -2, 0, 5

• -2 and 5 only

• -2, 2, and 8

• 2 and 8 only

Answer explanation

g'=f
g has an inflection point when f goes from increasing to decreasing or decreasing to increasing.

 $If\int_2^1f\left(x\right)dx=-3\ and\ \int_2^4f\left(x\right)dx=2.\ then\ \int_1^4f\left(x\right)dx=?$  

• 1

• -5

• -1

• 5

Answer explanation

Remember that you can change the order of the limits by putting a negative out front of the integral.

The graph of the function f is shown. At which of the following points is f'(x)<0 and f"(x)>0?

• c

• a

• b

• d

Answer explanation

f' shows where slopes are positive and negative

f" shows where a graph is concave up or down

Selected values from the function f(x) are shown in the table. When a left Riemann sum with two subintervals is used to approximate $\int_0^5f\left(x\right)dx$  , the value is...

• 16

• 21

• 15

• 11

For a particle moving along the x-axis, x (1 )=5 and v (1)= 2. At time t = 1, it can be said that the particle is...

• Speeding up

• moving toward the origin

• moving away from the origin

• slowing down.

Selected values of the twice differentiable function f and its derivative are shown in the table. What is the average value of f' from x = 2 to x = 8?

• 16

• 1/3

• 2

• 8/3

Answer explanation

f(b)-f(a)/b-a

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6

 $\frac{d}{dx}\left(e^{\frac{x}{5}}\right)=?$  

•  $-5e^{\frac{x}{5}}$  

•  $\frac{1}{5}e^{\frac{x}{5}}$  

•  $e^{\frac{x}{5}}$  

•  $5e^{\frac{x}{5}}$  

Answer explanation

The derivative of e^x is e^x, BUT you have to do chain rule on the exponent if it is other than a x.

Tags

CCSS.HSF-IF.C.8B