Math 241 Week 7 Midterm Review

Random key concepts

100

If at the critical point x=2, f'' is positive, is this a maximum or minimum value?

Minimum

200

What are two other ways to say "find the derivative"?

Find the instantaneous rate of change

Find the slope at a point

300

Where are the three places where a function could be discontinuous?

1. holes 2. asymptotes 3. breaks

400

If a numerator and a denominator can be simplified and cancel each other out, what will be at that x-value on the graph?

A hole

Critical points; min/max

100

Where do you find critical values?

Where f' = 0 or f' = undefined

200

The first derivative goes from "what" to "what" at a maximum?

Positive to Negative

300

Find the critical value(s) of f(x) = x²-4x-12

f'(x) = 2x - 4

0 = 2x - 4

4 = 2x

x = 2

400

Find the critical values of g(n) = n³-3n²-9n

Then determine the intervals of increasing and decreasing

n = 3 and n = -1

Increasing: (-inf, -1) U (3, inf)

Decreasing: (-1,3)

Optimization

100

If the domain of a function is (-infinity, infinity), will the function have any absolute extrema?

No

200

Where are the two places you will find a global maximum or minimum?

At critical points and endpoints.

300

If revenue is given by the function R(p)=1700p-10p^2, what price would tickets have to be sold at to maximize revenue?

$85

400

A manufacturer can produce sunglasses at a cost of $5 apiece and estimates that if they are sold for x dollars apiece, consumers will buy 100(20 - x) sunglasses a day. At what price should the manufacturer sell the sunglasses to maximize profit?

P = R - C

P = -100x^2+2500x-1000

price for maximized profit = 12.5 dollars

Linear approximation and marginal analysis

100

If the second derivative is POSITIVE, the linear approximation is... (overestimate/underestimate) - Draw a graph to show your thinking

Underestimate

200

If the second derivative is NEGATIVE, the linear approximation is... (overestimate/underestimate) - Draw a graph to show your thinking

Overestimate

300

What is marginal analysis? What does it show us?

A way of approximating the change associated with increasing production by ONE unit. 

C(x+1) - C(x) = Cost to produce one more item

R(x+1) - R(x) = Revenue from selling one more item 

P(x+1) - P(x) = Profit from selling one more item

400

What is the estimated total profit in selling 101 items if P(100) = 10,150 and P'(100) = 89?

Total profit in producing 100 items = 10,150

Approx cost to produce the 101st item = 89

Total cost to produce 101 items = 10,239