AP Calc AB Review Flashcards

tanθ=

sinθ/cosθ

cotθ=

cosθ/sinθ

cscθ=

1/sinθ

secθ=

1/cosθ

cotθ=

1/tanθ

sin²θ + cos²θ=

1

tan²θ +

cot²θ + 1=

csc²θ

sin(2θ)=

2sinθcosθ

cos(2θ)=

cos²θ - sin²θ

2cos²θ - 1

1-2sin²θ

cos²θ=

(1+cos(2θ))/2

sin²θ=

(1-cos(2θ))/2

sin(-θ)=

-sinθ

cos(-θ)=

cosθ

tan(-θ)=

-tanθ

∫a dx=

ax + c

∫xⁿ dx=

(xⁿ⁺¹/(n+1)) + c

∫1/x dx

In|x|+c

∫e^x dx=

e^x + c

∫a^x dx=

(a^x / Ina) + c

∫Inx dx=

xInx-x+c

∫sinx dx=

-cosx + c

∫cosx dx=

sinx + c

∫tanx dx=

In|secx|+c

∫cotx dx=

In|sinx|+c

∫secx dx=

In|secx + tanx| +c

∫cscx dx=

-In|cscx-cotx| + c

∫sec²x dx=

tanx + c

∫secxtanx dx=

secx+c

∫csc²x dx=

-cotx + c

∫cscxcotx dx=

-cscx + c

∫tan²x dx=

tanx-x+c

∫1/(a² + x²) dx

1/a tan^-1(x/a) + c

∫1/(√(a² - x²) dx=

sin^-1(x/a) +c

∫1/x√(x² - a²) dx=

1/a sec^-1|x/a| +c

Limit defintion of a derivative

• f'(x) = lim(h→0) [f(x+h) - f(x)] / h

• f'(a) = lim(h→0) [f(a+h) - f(a)] / h

Product Rule:

d/dx (fg)=

(f)(dg/dx) +(g)(df/dx)

Quotient Rule:

d/dx (Hi/Lo)=

Lo dHi-Hi dLo

/ (Lo)²

Chain Rule:

(f●g)’(x)=

or

dy/dx=

f’(g(x))●g’(x)

dy/du ● du/dx

Power Rules:

d/dx (xⁿ)=

nx^n-1

Derivative of a constant

=0

d/dx (a^u)=

In(a)a^uu’

d/dx(log_au)=

1/In(a) ● u’/u

d/dx (e^u)=

du/dx e^u

d/dx (Inu)=

u’/u

Inverse function:

d/dx (f^-1(x))=

1/(f^’(f^-1(x))

Intermediate value theorem

If a and b are any two points in an interval on which f is continuous, then f takes on every value between f(a) and f(b)

Intermediate value theorem for derivatives

If a and b are any two points in an interval on which f is differentiable, then the derivative f’ takes on every value between f’(a) and f’(b)

Average rate of change of a function f on [a,b]

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

Instantaneous rate of change of a function of f at x=a

f'(a) = lim(h→0) [f(a+h) - f(a)] / h

or

f'(a) = lim(x→a) [f(x) - f(a)] / (x - a)

mean value theorem (for derivatives)

If y=f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b), then there is at least one point c in (a,b) such that f’(c) = (f(b) - f(a)) / (b - a)

Average value of a function on a interval [a,b]

Average value of f(x) on [a,b] = (1/(b - a)) ∫[a to b] f(x) dx

Mean value theorem (for definite integrals)

If f is continuous on [a,b], then at some point c [a,b], f(c) = (1/(b - a)) ∫[a to b] f(x) dx

(at some point, the function equals its average value on the interval)

Fundamental Theorem of Calculus

d/dx ∫[a to x] f(t) dt = f(x)

or

d/dx ∫[a to u] f(t)dt = u’f(u)

and

∫[a to b] f(x)= F(b)-F(a)

[F(x) is an antiderivative of f(x)]

∫[a to b] f(x)dx=

-∫[a to b] f(x)dx

∫[a to a] f(x) dx=

0

∫[a to b] k● f(x)dx=

k ∫[a to b] f(x) dx

∫[a to b] -f(x)dx=

-∫[a to b] f(x)dx

∫[a to b] f(x) (+-) g(x) dx=

∫[b to a] f(x) dx (+-) ∫[b to a]g(x)dx

∫[a to c] f(x)dx=

∫[a to b] f(x)dx +∫[b to c] f(x)dx

Improper integrals, f(x) continuous on [a, infinity]

∫[a to infinity] f(x)dx= lim b→infinity ∫[a to b] f(x)dx

L’Hopital’s rule for indeterminate

forms states that if a limit yields a form like 0/0 or ∞/∞, then you can take the derivative of the numerator and denominator to evaluate the limit.

exponential growth equation

y=y_oe^kt

Linerarization

If f is differentiable at x=a, then the equation of the tangent line, L(x)=f(a)+f’(a)(x-a), defines the linearization of f at a. The standard linear approximation of f at a is f(x)≈L(x)