Math Pop Quiz NoteCards

For them pop quizzes

Definition of Continuity

See Picture

Extreme Value Theorm

If f(x) is continuous on the closed interval \[a,b\] there is at least one min and max

Critical Points

When f(x) = 0 or is undefined

Global Min and Max

Lowest and highest value in the interval

Point of Inflection

When f”(x) = 0 or is undefined and there is a sign change

Definition of a Derivative

LIM h→0 f(x+h)-f(x) / h

Product Rule

VdU + U\*dV

Quotient Rule

VdU - U\*dV / V^2

d/dx of x^n

n \* x ^ n-1

d/dx ln(u)

u’/u

d/dx e^u

e^u \* u’

d/dx sin^-1 (u)

u’ / sqrt(1-u^2)

d/dx cos^-1 (u)

\-u’ / sqrt(1-u^2)

d/dx tan^-1 (u)

u’ / 1+u^2

d/dx cot^-1 (u)

\-u’ / 1+u^2

d/dx a^u

a^u \* ln(a) \* u’

d/dx (log_a U)

1/ln(a) \* u’/u

Intermediate Value Theorem

In a continuous interval \[a,b\] there is at least one number x=c in the open interval a,b such that f(c) = y

Mean Value Theorem

If a function f is continuous on the closed interval \[a,b\] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over \[a,b\].

Rolle’s Theorm

if a function f is continuous on the closed interval \[a, b\] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.

Fundamental theorem of calculus

See Picture

L’Hopitals Rule

whenever direct substitution of a limit yields an indeterminate form. This means that the limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.

Mean Value Theorm for Integrals

for a continuous function over a closed interval, there is a value c such that f(c) equals the average value of the function

Integration by Parts

Int(UdV) = VU - Int(V\*dU)

sin^2(o) + cos^2(o)

1

1 + tan^2(o)

sec^2(o)

1+cot^2(o)

csc^2(o)

sin 2o

2 \* sin(o) \* cos(o)

cos 2o

cos^2 (o) - sin^2 (o)

2cos^2(o)-1

1-2sin^2(o)

tan 2o

2tan(o) / 1-tan^2(o)

sin^2(o)

(1-cos 2o) / 2

sin o/2

\+ or - sqrt(1-cos(o) / 2)

Cos^2(o)

(1 + cos 2o) / 2

Cos o/2

\+ or - sqrt(1+cos(o) / 2)

tan o/2

\+ or - sqrt ((1-cos(o))/(1+cos(o)))