calc 1 review

sinx right angle definition

opposite/ hypotenuse

cosθ right triangle definition

adjacent/hypotenuse

tanθ right triangle definition

opposite/adjacent

cscx right triangle definition

hypotenuse/opposite

secθ right triangle definition

hypotenuse/adjacent

cotθ right triangle definition

adjacent/opposite

domain of y=sin⁻¹x

$[-1, 1]$,

range of y=sin⁻¹x

[-π/2, π/2]

domain arccosx

[-1,1]

range arccosx

[0,π]

cscθ reciprocal identity

1/sinθ

secθ reciprocal identity

1/cosθ

of the trig functions, which are odd and even?

all trig functions are odd EXCEPT for:

• cosθ

• secθ

  • sec(-θ)=1/cosθ

• 
  

sin²θ + cos²θ =

1

tangent double angle formula

sin(2θ) double angle formula

sin(2θ)= 2sinθcosθ

cos(2θ) double angle formula(s)

double angle formulas

Double-angle formulas are trigonometric identities that show how to find the trig values of twice an angle (2θ) using the single angle (θ)

half angle formulas

used for finding the exact value of trig function for a value that is not on the unit circle (e.g finding 15º when 30º exists)

• sin(θ/2) and cos(θ/2) are the same ±√(1±cosθ)/2, only numerator of sin is 1-cosθ

power reducing formula tan

power reducing formula sin and cos

tangent half angle formulas

tan²θ + 1 =

sec²θ

1 + cot²θ =

csc²θ

chain rule

used for differentiating composite functions (functions within functions). If you have a function y=f(g(x)), the chain rule states:

dx/dy​=f ′ (g(x))⋅g ′ (x)

• Derivative of the outer function (evaluated at the inner function) multiplied by the derivative of the inner function.

power rule

Differentiating functions of the form f(x)=xⁿ, where n is a real number.

differentiating a constant

0

because no change shawty

product rule

used for differentiating products of two functions.

quotient rule

use for Differentiating ratios of two functions

base e exponential derivative rule

essentially the chain rule

non base e exponential derivative rule

derivative of a logarithm rule

derivative of sinx

cosx

derivative of cosx

-sinx

derivative of tanx

sec²x

derivative of cotx

-csc²x

derivative of secx

secxtanx

derivative of cscx

-cscxcotx

domain of arctan

all real numbers

(-∞,∞)

range arctan

(-π/2,π/2)

• NOT INCLUDING**

derivative of arctan

• derivative of arccot is the same thing but negative -1/(1+x²)

derivative of arcsin

• arccos is the same thing but negative -u’/√1-u²

derivative of arcsec

• arccsc is the same but negative

how many antiderivatives of a given function are there

infinetely many

antiderivative

a mathematical function that reverses the process of differentiation. If you take the derivative of a function and get a specific result, the antiderivative is the original function you started with.

• When we write F(x)+C, we denote the entire family of antiderivatives of f.

∫1/x dx=

ln|x|+C

power rule of integration

∫e^x dx=

e^x+C

∫a^xdx=

axln(a)+C

∫cos(x)dx=

sin(x)+C

∫sin(x)dx=

−cos(x)+C

∫sec2(x)dx=

tan(x)+C

∫csc2(x)dx=

−cot(x)+C

∫sec(x)tan(x)dx=

sec(x)+C

∫csc(x)cot(x)dx=

−csc(x)+C

∫1/x²+1 dx=

arctan(x)+C

∫1/√1-x² dx=

arcsin(x)+C

How do you integrate a constant times a function?

move it before the integral symbol then integrate

can you integrate a factor x factor?

NOO