AP Calc AB exam review

d/dx (x)

1

d/dx sin(x)

cos(x)

d/dx cos(x)

\-sin(x)

d/dx tan(x)

sec^2(x)

d/dx sec(x)

sec(x)tan(x)

d/dx csc(x)

\-csc(x)cot(x)

d/dx cot(x)

\-csc^2(x)

d/dx sin^-1(x)

1/sqrt(1-x^2)

d/dx cos^-1(x)

\-1/sqrt(1-x^2)

d/dx tan^-1(x)

1/(1+x^2)

d/dx a^x

a^x\*ln(a)

d/dx e^x

e^x

d/dx ln(x)

1/x

d/dx mx

m

d/dx c

0

d/dx x^n

n\*x^(n-1)

d/dx sqrt(x)

1/2sqrt(x)

∫ m dx

mx+c

∫ x dx

x^2/2

∫ x^n dx

(1/n+1)x^(n+1)+c

∫ x^-1 dx

ln|x|+c

∫ 1/(ax+b) dx

(1/a)ln|ax+b|+c

∫ e^x dx

e^x+c

∫ cos(x) dx

sin(x)+c

∫ sin(x) dx

\-cos(x)+c

∫ sec^2(x) dx

tan(x)+c

∫ sec(x)tan(x) dx

sec(x)+c

∫ csc(x)cot(x) dx

\-csc(x)+c

∫ csc^2(x) dx

\-cot(x)+c

∫ a^x dx

(a^x/ln(a))+c

∫ sqrt(x) dx

(2/3)x^(3/2)+c

slope of line tangent to y=f(x) at x=a

m = f’(a)

tan line formula at x=a

y=f(a)+f´(a)(x-a)

(cf´(x))

cf´(x)

(f+/- g)´

f´(x)+/-g´(x)

product rule (fg)´

f´g+fg´

quotient rule (f/g)´

lo\**dhi-hi**dlow/lolo

power rule (x^n)´

(n)x^(n-1)

chain rule (f(g(x)))

f´(g(x))g´(x)

sin(0)

0

cos(0)

1

sin(π/6)

1/2

cos(π/6)

sqrt(3)/2

sin(π/4)

sqrt(2)/2

cos(π/4)

sqrt(2)/2

sin(π/3)

sqrt(3)/2

cos(π/3)

1/2

sin(π/2)

1

cos(π/2)

0

sin(2π/3)

sqrt(3)/2

cos(2π/3)

\-1/2

sin(3π/4)

sqrt(2)/2

cos(3π/4)

\-sqrt(2)/2

sin(5π/6)

1/2

cos(5π/6)

\-sqrt(3)/2

sin(π)

0

cos(π)

1

sin(7π/6)

\-1/2

cos(7π/6)

\-sqrt(3)/2

sin(5π/4)

\-sqrt(2)/2

cos(5π/4)

\-sqrt(2)/2

sin(4π/3)

\-sqrt(3)/2

cos(4π/3)

\-1/2

sin(3π/2)

\-1

cos(3π/2)

0

sin(5π/3)

\-sqrt(3)/2

cos(5π/3)

1/2

sin(7π/4)

\-sqrt(2)/2

cos(7π/4)

sqrt(2)/2

sin(11π/6)

\-1/2

cos(11π/6)

sqrt(3)/2

AROC formula

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

IROC formula

(f(a+h)-f(a))/h

tan line overestimates if

f is concave down, f´ is decreasing, f´´ is negative

tan line underestimates if

f is concave up, f´ is increasing, f´´ is positive

average value formula

1/(b-a)∫ a to b f(x)dx

mean value theorem

if f(x) is continuous and differentiable on the open interval (a,b) then there is a number c so f´(c)=(f(b)-f(a))/(b-a)

the slope of the tan line and sec line are equal

rel min in f

f´ sign change - to +, f´´ is positive

rel max in f

f´ sign change + to -, f´´ is negative

f is increasing

f´ is positive

f is decreasing

f´ is negative

POI (concavity change) in f

f´ has rel extrema, f´´ changes sign

f is concave up

f´ is increasing, f´´ is positive

f is concave down

f´ is decreasing, f´´ is negative

f has saddle point

f´=0, f´´=0 or DNE

f has cusp

f´ DNE, f´´ DNE

LRAM underestimates when

f is increasing, f´ is positive

LRAM overestimates when

f is decreasing, f´ is negative

RRAM overestimates when

f is increasing, f´ is positive

RRAM underestimates when

f is decreasing, f´ is negative

trapezoidal sums underestimate when

f is concave down, f´ is decreasing, f´´ is negative

trapezoidal sums overestimate when

f is concave up, f´ is increasing, f´´ is positive

position

x(t), location of an object at time t

velocity

v(t), rate of change in position

acceleration

a(t), rate of change in velocity

derivative of position

velocity x´(t)=v(t)

derivative of velocity

acceleration v´(t)=a(t)

2nd derivative of position

acceleration x´´(t)=a(t)

antiderivative of acceleration

total change in velocity ∫a(t)dt=v(t)+c

antiderivative of velocity

total change in position ∫v(t)dt=x(t)+c