Topic 5: Calculus

Limit of a function:

• Limits describe the output of a function as the input approaches a certain value.

• If f(x) approaches a real number (L) as x gets closer to (but not equal to) a real value, a, (for both x<a and x>a), then we say that the limit of f(x) as x approaches a is L.

We write this as lim(x→a) f(x) = L

• As x approaches a → we can say the limit exists because it is what we believe to be at the "hole" in the graph.

• This "hole" in the graph does not exist (it is a point where there is no value), but you can estimate what it would be.

*Note: f(a) may or may not be doable.

In this graph, following the function, it converges to f(x)= 4 whether you follow it from the right or left.

Can f(a) be evaluated?:

• Consider f(x) and g(x) - They are the same

  • g(0) can be calculated, it's 5

  • f(0) can not be calculated (divide by 0)

    • The function has a hole at x=0

    • Point of Discontinuity

  • As x approaches 0 from either the right or left, the function approaches 5 from either above or below

  • Limit of f(x) is 5, as x approaches 0

Left and Right Approach:

• Think of the number line in terms of right or left

• x → a+ means "x tends to a from the right"

  • start at values greater and decrease towards a

• x → a- means "x tends to a from the left"

  • start at values lower and increase towards a

• It is possible they don't match up

The limit only exists if both limits (from both directions) equal the same value.

Examples:

1. limx→5 (x-1)(x+5)

• Answer: (5-1)(5+5) = 4·10=40

2. limx→2 (3(x^2-1))/x-1

3. limx→1 (x^2-1)/x-1

4. limx→0 (sin(x))/x

Limits to Infinite:

• Important to understand functions as variable get really big.

  • Limit that x approaches ±∞

• Dealing with limits to infinity we first identify 3 basic properties to the right

• If an expression has polynomials in both the numerator and the denominator

  • Divide every term by the highest order term in x

    • See then which terms go to infinity, a constant, or zero.

Example:

Derivative Function:

Instantaneous Rate

Secant to Tangent

Example

Power Rule

Special Derivative Rules

Example

Tangents and Normals

Examples

The Chain Rule:

Example:

The Product Rule:

Example:

The Quotient Rule:

Example:

Intervals:

How to check for Increasing/Decreasing:

Example:

Turning Points (First Derivative Test):

Easy Reference Guide:

Example:

Second Derivative:

Concavity:

Inflection Point:

Second Derivative Test:

Summary fo Inflection Points:

Example:

Sketches:

Example:

Starting from f'(x):

Example:

Optimization

Introduction:

Optimization Process:

Examples:

Kinematics

Displacement:

Velocity:

Acceleration:

Examples: