Integration and Differentiation

Derivative Formulas

100

d/dx [sin(x)]

cos(x)

200

d/dx [cos(x)]

-sin(x)

300

d/dx [csc(x)]

-csc(x)cot(x)

400

d/dx [arcsin(x)]

1/(sqrt(1-x²)

500

d/dx [arccsc(u)]

u'/|u|sqrt[(u²)-1]

Integration Formulas

100

int. of cos(x)dx

sin(x) + C

200

int. of sin(x)dx

-cos(x) + C

300

int. sec²xdx

tanx + C

400

int. of [dx/(1+x²)]

arctan(x) + C

500

int. of a^xdx

(a^x)/lna + C

Misc

100

What are the prerequisite of AP Calculus BC

Honors Algebra II, Honors Pre-­Calculus, and AP Calculus AB

200

d/dx (lnx)

1/x

300

If a student’s stress level S(t) is increasing at a rate of S′(t)=5e^t as the AP exam approaches, how fast is their stress growing when they realize they studied the wrong unit?

Exponentially. Like, really exponentially.

400

If you watch Netflix at a rate of f(t)=2t episodes/hour over a 4-hour period, how much of your soul have you lost?

16 episodes. And your dignity.

500

According to the Extreme Value Theorem, if your GPA is continuous on a closed interval [9th,12th], it must have a maximum and minimum. When did it reach the minimum?

Right after you saw your Calc BC final grade.

FTC Application

100

What does FTC stand for?

Fundamental Theorem of Calculus 

200

int. from [-6,6] of f'(t)dt

f(6)-f(-6)

300

f(x)=integral [0,x] cos(t)dt, then f′(x)=

cos(x)

400

d/dx [int. [1,x²] ln(t)dt] 

2xln(x²)

500

int. from [0,3] of (-x³+3x²-2)dx

0.75

Problems

100

d/dx|x=pi/2 [cos(x)]

-1

200

int. (x³+3x²+5) dx

x⁴/4+x³+5x+C

300

d/dx x³tanx

3x²tan(x) + x³sec²x 

400

int. of cos(x)sin⁵(x)

sin⁶(x)/6 + C

500

int. of xlnxdx

1/2(x²lnx)-1/4(x²)+C