AP Calc AB Semester 1

Intermediate Value Theorem

If f(x) is a continuous function on [a,b], then f(x) will take on all values between f(a) and f(b).

Slope

Average Rate of Change

Derivative

Instantaneous Rate of Change, or the Slope at a Point

Limit definition of a derivative

lim_h→0 (f(x+h) - f(x))/((x+h)-x)

d/dx (C)

0

d/dx (C * f(x))

C * f’(x)

d/dx (x^C)

C*x^C-1

d/dx (f(x) ± g(x))

f’(x) ± g’(x)

d/dx (f(x) * g(x))

f(x)*g’(x) + g(x)*f’(x)

d/dx (f(x)/g(x))

(g(x)*f’(x) - f(x)*g’(x))/((g(x))²)

or

LodHi - HidLo/Lo²

sin²(x) + cos²(x)

1

sin(A+B)

sinA*cosB + sinB*cosA

cos(A+B)

cosA*cosB - sinA*sinB

continuity

lim_x→A⁻f(x) = lim_x→A⁺f(x) = f(A)

lim_x→C (f(x) ± g(x))

lim_x→C f(x) ± lim_x→C g(x)

lim_x→C K

(K is another constant)

K

lim_x→C (K*f(x))

K * lim_x→C (f(x))

lim_x→C (f(x) * g(x))

lim_x→C f(x) * lim_x→C g(x)

lim_x→C (f(x) / g(x))

lim_x→C f(x) / lim_x→C g(x)

particle goes up

velocity is positive

acceleration is negative (going up more slowly, think Earth’s gravity)

particle goes down

velocity is negative

acceleration is negative

particle changes direction

velocity changes sign

particle speeds up

velocity and acceleration have the same sign

+/+ or -/-

Particle slows down

velocity and acceleration have opposite signs

+/- or -/+

d/dx sin(x)

cos(x)

d/dx cos(x)

-sin(x)

d/dx tan(x)

sec²(x)

d/dx sec(x)

sec(x)*tan(x)

d/dx csc(x)

-cot(x)*csc(x)

d/dx cot(x)

-csc²(x)

d/dx sin^-1(x)

1/√(1-x²)

d/dx cos^-1(x)

-1/√(1-x²)

d/dx tan^-1(x)

1/(x²+1)

d/dx cot^-1(x)

1/(x²+1)

d/dx sec^-1(x)

1/(|x|*√(x²-1))

d/dx csc^-1(x)

-1/(|x|*√(x²-1))

Extreme Value Theorem

if f is continuous on a closed interval [a,b], then f has a minimum and a maximum within the interval

Maximum

f’(x) changes from + → -

f’’(x) is +

Minimum

f’(x) changes from - → +

f’’(x) is -

Concave Up :)

MAKES MINIMUMS

f’’(x) is +

Concave Down :(

MAKES MAXIMUMS

f’’(x) is -

Point of Inflection

f’(x) has a (relative) maximum or minimum

f’’(x) changes sign

d/dx e^x

e^x

d/dx ln(x)

1/x

log_c1

0

log_cc

1

c^log_c(x)

x

log_c(m*n)

log_c(m) + log_c(n)

log_c(m/n)

log_c(m) - log_c(n)

log_c(mⁿ)

n*log_c(m)

log_c(m)

log_n(m)/log_n(c)