MA 165 Exam 1

Product Rule

f’(x)g(x)+f(x)g’(x)

Quotient Rule

(LoDeHi-HiDeLo)?Lo²

Power Rule

X^n= nx^n-1

Log Power Rule

ln(a^b)=b*ln(a)

Domain and Range of Sin-1(x)

D: [-1,1], R:[-pi/2,pi/2]

Domain and range of cos-1(x)

D:[-1,1], R:[0,pi]

Domain and range of tan-1(x)

D:(- inf, + inf), R: (-pi/2, pi/2)

Complementary Inverse Trig Identity

sin-1(x)+cos-1(x)=pi/2 when x [-1,1]

Sin Double-Angle Formula

2sinxcosx

Cos Double-Angle Formula (sin only)

1-2sin²x

Cos Double-Angle Formula (cos only)

2cos²x-1

Cos Double-Angle Formula (cos and sin)

cos²(x)-sin²(x)

Pythagorean Trig Identities (sin and cos)

sin²x+cos²x=1

Pythagorean Trig Identities (tan and sec)

1+tan²x=sec²x

Pythagorean Trig Identities (cot and csc)

1+cot²x=csc²x

Condition for Function to have an inverse

function must be 1-to-1

Doman and Range Relationship of Inverse Functions

Domain(f^-1)=Range(f) and Range(f^-1)=Domain(f)

Derivative of an Inverse Function Formula

if f(a)=b, then (f^-1)b=1/f"‘(a)

Fundamental Trig Limit at 0

sin(kx)/(kx) = 1 and sin(ax)/(bx)= a/b

Squeeze Theorem Statement

you get the idea

Indeterminate Form (0/0) Strategy (Radicals)

Multiply numerator and denominator by conjugate

Horizontal Asymptotes and Limits at Infinity

y=L is a horizontal asymptote if lim as x approaches (+ or - infinite) of f(x) = L

n=d, coefficients

n>d, no H.A.

n<d, H.A. = y=0

Vertical Asymptotes Condition

set denominator equal to 0. x=a is a VA if the limit as x approaches (+ or - a) = (+ or - infinite)

IVT

if f is continuous from [a,b], and N is between f(a) and f(b), there exists at least one place between a and b where f of c = N

Limit Definition of the derivative f’(x)

f(x+h)-f(x) all over h

Relationship between Differentiability and Continuity

Differentiable = Continuous, but Continuous =/ Differentiable

Slope of Parallel vs. Perpendicular Lines

Parallel = same slope. Perpendicular = negative reciprocal

Point-Slope form for Tangent Line Equation

y-f(a)=f’(a)(x-a)

Limit at Infinite with Square Roots

Square root of x²=IxI= -x when x is less than 0 (aka as x approaches - infinite)

Conditions for Continuity at x=a

1. f(a) is defined

2. lim as x approaches a = f(x) exists

3. lim as x approaches a = f(a)

Right Triangle Method for Inverse Trig Algebraic Forms

set theta = trig-1(…), label a right triangle with sides and use Pythagoreans theorem.

Limit trick for derivative definition matching

Inverse Relationship between Exponential and Logarithmic Functions

y=x^b —> log base b of y = x. y=e^x —> ln(y)=x

Logarithm Product, Quotient, and Power Rules

Log(xy)=logx+logy. Log(x/y)=logx-logy. Log base b (x^k) = k log base b (x)

Limit as x approaches infinite of tan inverse x / x

0

Limit as x approaches - infinite of tan inverse x / x

0

Limit as x approaches 0 of tan inverse x / x

1

Lim as x approaches 0 of sinx/x

1

Lim as x approaches 0 of 1-cosx / x

0

derivative of sinx

cosx

derivative of cosx

-sinx

derivative of tanx

sec²x

derivative of secx

secxtanx

derivative of cotx

-csc²x

derivative of cscx

-cscxcotx