BackBusiness Calculus: Limits, Derivatives, and Applications Study Guide
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Limits and Continuity
Definition of a Limit
The limit of a function as x approaches a value describes the behavior of the function near that value. It is a foundational concept in calculus, used to define derivatives and continuity.
Limit Notation: represents the value that f(x) approaches as x gets arbitrarily close to a.
Example:
Definition of Continuity
A function f(x) is continuous at x = a if:
is defined
exists
If any of these conditions fail, the function is discontinuous at x = a.
Evaluating Limits from Graphs
To find , observe the value as x approaches a from the left.
To find , observe the value as x approaches a from the right.
If both one-sided limits are equal, the two-sided limit exists and equals that value.
If the function jumps or has a hole at x = a, the limit may not exist or may not equal f(a).
Derivatives and Their Applications
Limit Definition of the Derivative
The derivative of a function f(x) at a point x is defined as:
This measures the instantaneous rate of change of f(x) with respect to x.
It is the slope of the tangent line to the graph of f(x) at x.
Example: Derivative of
Using the limit definition:
Expand and simplify:
Average Rate of Change
The average rate of change of f(x) from x = a to x = b is:
This represents the slope of the secant line between (a, f(a)) and (b, f(b)).
Finding Tangent Lines
The equation of the tangent line to f(x) at x = a is:
Here, is the slope at x = a, and (a, f(a)) is the point of tangency.
Horizontal Tangents
Horizontal tangents occur where .
Set the derivative equal to zero and solve for x to find these points.
Applications to Business: Cost Functions
Total, Marginal, and Average Cost
Total Cost (C(x)): The total cost to produce x units.
Marginal Cost (C'(x)): The instantaneous rate of change of total cost with respect to x; approximates the cost of producing one more unit.
Average Cost (A(x)): The total cost per unit, .
Example: Cost Function (in thousands of dollars)
Total cost to produce 4,000 units:
(thousand dollars)
Marginal cost at 4,000 units:
(thousand dollars per thousand units)
Average cost at 4,000 units:
(thousand dollars per thousand units)
Rate of change of average cost at 4,000 units:
(thousand dollars per one thousand units)
Summary Table: Types of Discontinuities
Type | Description | Example |
|---|---|---|
Removable | Hole in the graph; limit exists but f(a) is undefined or not equal to the limit | at x = 1 |
Jump | Left and right limits exist but are not equal | Piecewise function with different values at a point |
Infinite | Function approaches infinity at a point | at x = 0 |
Practice Problems and Solutions
Find the limit: Solution: Factor numerator: , so limit is as , answer is 3.
Find the derivative using the limit definition: Solution:
Find the average rate of change from x = 1 to x = 1: Solution:
Find the tangent line at x = 1: Solution: , , so
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