Topic 5: Calculus
Chapter 5: Differential Calculus
5.1 Limits and Convergence
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:
limx→5 (x-1)(x+5)
Answer: (5-1)(5+5) = 4·10=40
limx→2 (3(x^2-1))/x-1
limx→1 (x^2-1)/x-1
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:
5.2 The Derivative Function
Derivative Function:
Instantaneous Rate
Secant to Tangent
Example
Power Rule
Special Derivative Rules
Example
Tangents and Normals
Examples
5.3 Differentiation Rules
The Chain Rule:
Example:
The Product Rule:
Example:
The Quotient Rule:
Example:
5.4 Graphical Interpretations of First and Second Derivatives
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:
5.5 Applications of Differential Calculus: Optimization and Kinematics
Optimization
Introduction:
Optimization Process:
Examples:
Kinematics
Displacement:
Velocity:
Acceleration:
Examples: