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Mathematics for Business Decisions, part II. Differentiation. Math 115b. Ekstrom Math 115b.

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## Differentiation

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Differentiation, Part I

Properties of Graphs

- Describe the graph:
- Where is the function…
- increasing?
- decreasing?
- decreasing the fastest?

Ekstrom Math 115b

Differentiation, Part I

Properties, cont.

- Describe f(x).

Where is f:

- positive?
- negative?
- zero?
- increasing?
- decreasing?

Ekstrom Math 115b

Differentiation, Part I

Rate of Change

- Rate of change of a linear function is called “slope”
- Denoted as m in y = mx + b
- How is it defined?
- What if the function is not linear?

Ekstrom Math 115b

Differentiation, Part I

Rate of Change, cont.

- Consider the function from earlier:
- Can we define a “slope” of this line?

Ekstrom Math 115b

Differentiation, Part I

Example Data

- Consider the following set of data points (Tucson temperatures before, during, and after a thunderstorm):

Ekstrom Math 115b

Differentiation, Part I

Example, cont.

- Perhaps plotting the data will give us a better description:
- What is the rate of change of the temperature at 4:29 (16:29)?

Ekstrom Math 115b

Differentiation, Part I

Finding the “slope” at a point

- So what do we want to do?
- To evaluate the rate of change (slope) of f (x) at x, we should find the slope between the points before and after the point in question:

for some h.

Ekstrom Math 115b

Differentiation, Part I

Slope at a point

- As h gets smaller and smaller, the approximation of the slope gets better and better.
- The derivative of a function is slope of a tangent line at a point on any curve, and can be calculated by:
- It is usually denoted as or

Ekstrom Math 115b

Differentiation, Part I

Algebra Review

- What does f (x + h) mean?
- Ex.
- Soln:
- It means you evaluate the function at the quantity, x + h. Do NOT simply add h to f(x)! This will ultimately lead to a slope of 1.

Ekstrom Math 115b

Differentiation, Part I

Algebra Review, cont.

- Example: Calculate the derivative of the function f (x) = 5x + 2 using the difference quotient.
- Solution:

Surprised?

Ekstrom Math 115b

Differentiation, Part I

Example calculations

- Calculate the derivatives of the following functions:

Ekstrom Math 115b

Differentiation, Part I

Difference Quotient

- The derivative of a function is the slope of the line tangent to any point on the curve, f (x).
- It is calculated by finding the limit:
- This gives an instantaneous rate of change of the function, f (x).

Ekstrom Math 115b

Differentiation, Part I

Instantaneous Rate of Change

- What do we mean by instantaneous?
- If h was one unit, and we calculated the difference quotient, then we would be finding the average rate of change between the points before and after the point in question.
- We want h to be smaller and smaller (closer and closer to 0) so that the length 2h is approximately 0 so our quotient will stabilize.

Ekstrom Math 115b

Differentiation, Part I

Tangent Line

To visualize the tangent line, think of a bird’s eye view of a curvy road at night. The headlights of a car traveling along this road will not follow the curves of the pavement. The path of the headlights represents the tangent line to the curvy road.

Ekstrom Math 115b

Differentiation, Part I

Tangent Line

- The equation of the tangent line should be y = mx+ b
- Slope of tangent line is equal to the derivative at every point x
- m = f ’(x), where m is the slope of the tangent line
- Since we know the slope and a point on the line, we can find the equation of the tangent line
- If the derivative at the point exists, then the tangent line to the graph of f at the point (a, f (a)) has the equation

Ekstrom Math 115b

Differentiation, Part I

First Example

- Find the slope of the line tangent to the graph of

at the point (3, f (3)).

- Find an equation for the tangent line at that point.

Ekstrom Math 115b

Differentiation, Part I

Second Example

- Let f (x) = x3 + 6
- Find the equation of the line tangent to f (x) at the point (-1, f (-1)).
- Luckily, you don’t have to do this by hand every time
- Differentiating.xls

Ekstrom Math 115b

Differentiation, Part I

Graphing the Derivative

- Want to get a sketch of the derivative graph
- Interpretation of derivative is slope of tangent line
- What does an ordered pair represent on the derivative graph?
- How can you obtain the ordered pairs?

Ekstrom Math 115b

Differentiation, Part I

Algebraic Rules

- If f (x) = k (constant), then f (x) = 0
- If f (x) is linear, f (x) = mx + b, and f (x) = m
- Why?
- If f (x) = a g(x), then f (x) =a g(x)
- If f (x) = g(x) h(x), then f (x) = g(x) h(x)
- Specifically, since P(q) = R(q) - C(q), then P(q) = R(q) – C (q) ANDP(q) = 0 when R(q) = C (q)

Ekstrom Math 115b

Derivitive Rules

Power Rule

f’ (xn) = nxn-1

Product Rule

f’(u∙v) = u ∙ f’(v) + v ∙ f’(u)

Quotient Rule

f’ (u/v) = v ∙ f’(u) – u ∙ f’(v)

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