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Squeeze Theorem

f is continuous at x=c if...

Intermediate Value Theorem

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

Global Definition of a Derivative

Alternative Definition of a Derivative

f '(x) is the limit of the following difference quotient as x approaches c

nx^(n-1)

1

cf'(x)

f'(x)+g'(x)

f'(x)-g'(x)

cos(x)

-sin(x)

sec²(x)

-csc²(x)

sec(x)tan(x)

dy/dx

f'(g(x))g'(x)

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

Critical Number

If f'(c)=0 or does not exist, and c is in the domain of f, then c is a critical number. (Derivative is 0 or undefined)

Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

Mean Value Theorem

The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.

First Derivative Test for local extrema

Point of inflection at x=k

Combo Test for local extrema

If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c.

If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.

If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.

Horizontal Asymptote

x+c

sin(x)+C

-cos(x)+C

tan(x)+C

-cot(x)+C

sec(x)+C

-csc(x)+C

Fundamental Theorem of Calculus #1

The definite integral of a rate of change is the total change in the original function.

Fundamental Theorem of Calculus #2

Mean Value Theorem for integrals or the average value of a functions

ln(x)+C

-ln(cosx)+C = ln(secx)+C

hint: tanu = sinu/cosu

ln(sinx)+C = -ln(cscx)+C

ln(secx+tanx)+C = -ln(secx-tanx)+C

ln(cscx+cotx)+C = -ln(cscx-cotx)+C

If f and g are inverses of each other, g'(x)

Area under a curve

Formula for Disk Method

Axis of rotation is a boundary of the region.

Formula for Washer Method

Axis of rotation is not a boundary of the region.

Inverse Secant Antiderivative

Inverse Tangent Antiderivative

Inverse Sine Antiderivative

Derivative of eⁿ

ln(a)*aⁿ+C

Derivative of ln(u)

Antiderivative of f(x) from [a,b]

Opposite Antiderivatives

Antiderivative of xⁿ

Adding or subtracting antiderivatives

Constants in integrals

Identity function

D: (-∞,+∞)

R: (-∞,+∞)

R: (-∞,+∞)

Squaring function

D: (-∞,+∞)

R: (o,+∞)

R: (o,+∞)

Cubing function

D: (-∞,+∞)

R: (-∞,+∞)

R: (-∞,+∞)

Reciprocal function

D: (-∞,+∞) x can't be zero

R: (-∞,+∞) y can't be zero

R: (-∞,+∞) y can't be zero

Square root function

D: (0,+∞)

R: (0,+∞)

R: (0,+∞)

Exponential function

D: (-∞,+∞)

R: (0,+∞)

R: (0,+∞)

Natural log function

D: (0,+∞)

R: (-∞,+∞)

R: (-∞,+∞)

Sine function

D: (-∞,+∞)

R: [-1,1]

R: [-1,1]

Cosine function

D: (-∞,+∞)

R: [-1,1]

R: [-1,1]

Absolute value function

D: (-∞,+∞)

R: [0,+∞)

R: [0,+∞)

Given f(x):

Is f continuous @ C

Is f' continuous @ C

Is f continuous @ C

Is f' continuous @ C

Yes lim+=lim-=f(c)

No, f'(c) doesn't exist because of cusp

No, f'(c) doesn't exist because of cusp

Given f'(x):

Is f continuous @ c?

Is there an inflection point on f @ C?

Is f continuous @ c?

Is there an inflection point on f @ C?

This is a graph of f'(x). Since f'(C) exists, differentiability implies continuouity, so Yes.

Yes f' decreases on X<C so f''<0

f' increases on X>C so f''>0

A point of inflection happens on a sign change at f''

Yes f' decreases on X<C so f''<0

f' increases on X>C so f''>0

A point of inflection happens on a sign change at f''