Calculus III

Theorems

Extreme Value Theorem

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If a function f is continuous on a finite closed interval [a,b], then f has both maximum and an absulute minimum.

Intermediate Value Theorem

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If a funtion f is continuous on[a,b] and k is a number such that f(a)<k<f(b), then there exists a number c in[a,b]such that f(c)=k.

Squeeze Theorem

Let the above condition be true.

Let the above condition be true.

If the limits of g(x) and h(x) as x tends to c are the same, value L.

If the limits of g(x) and h(x) as x tends to c are the same, value L.

By squeeze theorem, the limit of f(x) as x tends to c will also be L.

By squeeze theorem, the limit of f(x) as x tends to c will also be L.

Mean Value Theorem

Rolles's Theorem

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Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a)=0 and f(b)=0, then there is at least one point c in (a,b) such that f'(c)=0

A special condition of MVT

Applications of The Theorems

EVT

IVT

Squeeze Theorem

a

MVT

a

Rolles's Theorem

a

Parametric Functions

Finding the Derivative

1st

2nd

Converting to Cartesian Form

Arc Lengh of a Curve

y=f(x)

y=f(t), x=f(t)

Tangent Lines to Parametric Curves

horizontal tangent line

vertical tangent line

singular points

Polar Functions

Conversion with Cartesian equations

x=rcosθ, y=rsinθ

Finding the Derivative

Tangent Lines to Polar Curves

Arc Length of a Polar Curve

Vector Functions

Finding the Derivative

1st

2nd

Tangent and Normal Vectors

Tangent Vector

Normal Vector

Limits and Continuity

Limit

2 sides limit

Continuity

Continuous at x=c

f(c) is defined

exists

exists

Continuous on [a,b]

f is continuous on(a,b)

f is continuous from the right at a

f is continuous from the left at b

Derivatives

Definition

Differentiability

Relationship between continuity and differentiability

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If f(x) is differentiable in (a.b), f(x)is also continuous in (a.b). The reverse is not true.

A function is said to be differentiable at xo if the limit above exists

A function is said to be differentiable at xo if the limit above exists

Rules of Diferentiation

Constant multiples

Addition and subtraction

Higher derivatives

Chain Rule

Product Rule

Quatient Rule

Inverse funciton

Interpretation of f '

If f'(x)>0, f increasing

If f'(x)<0, f decreasing

If f'(x)=0, f constant

Interpretation of f "

If f ''(x)>0, f concave up

If f ''(x)<0, f concave down

If f ''(x)=0, f straight line

Critical Points

Relative extrema

f 'changes sign

Absolute Extrema

Theorem

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Suppose that f is continuous and has exactly one relative maximum(minimun) on an interval I,say at a then f(a) is the absolute maximum(minimum) of f on I.

Inflection Points

f ''=0 or undefined. Then f '' changes sign in the vicinity of those x-values.

First Derivative Test

Second Derivative Test

ε-δ definition of continuity

Definition

L'Hopital's Rule

Indeterminate form of type 0/0

Indeterminate form of type ∞/∞

Indeterminate form of type 0.∞

Indeterminate form of type ∞-∞

Applications of Derivatives

Related Rate

f'(t)

Maximum and Minimum

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First derivative.(Find the criticle points)

Tangent and Nomal Lines

Linear Approximations

for all points near a

for all points near a

Motion Along a Line

velocity

instantaneous speed

I v(t) I

acceleration