Derivative Rules
What are two other ways to say "find the derivative"?
Find the instantaneous rate of change
Find the slope at a point
What IS a critical point/value?
A critical value of a function f is a value of a so that a is in the domain of f and either f'(a) = 0 OR f'(a) is undefined
If the second derivative is POSITIVE, the linear approximation is... (overestimate/underestimate) - Draw a graph to show your thinking
Underestimate
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
Name the rule(s) you'd need to take the following derivative, then take it!
f(x) = (5x2-8x)3 + (32-10x)4
Chain Rule, Sum Rule, Power Rule, Constant Multiple Rule
3(10x-8)(5x2-8x)2 - 40(9-10x)3
The first derivative goes from "what" to "what" at a maximum?
positive to negative
If the second derivative is NEGATIVE, the linear approximation is... (overestimate/underestimate) - Draw a graph to show your thinking
Overestimate
If revenue is given by the function R(p)=1700p-10p2, what price would tickets have to be sold at to maximize revenue?
85
Name the rule(s) you'd need to take the following derivative, then take it!
f(x) =( (x7-4)/(x+2) ) - ( (x5-3x)/(4x+8) )
Difference Rule, Quotient Rule, Power rule, Constant Multiple rule
Find the critical value(s) of f(x) = x2-4x-12 ?
f'(x) = 2x - 4
0 = 2x - 4
4 = 2x
x = 2
Give a reason why linear approximation might be useful.
We may not have enough information to approximate a point on a function! This helps us approximate!
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
Name the rule(s) you'd need to take the following derivative, then take it!
( (x+8)/(x3-5) )5
Chain Rule, Quotient Rule, Power Rule
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)
Ethan's math t-shirt store sells shirts has an initial average cost of $25. Due to the major increase in the love for mathematics, his average cost is increasing by $5 per day. But, because the midterm is coming up, the growth rate of his cost is decreasing.
Express these 3 claims regarding the average cost of Ethan's t-shirts as a function of time!
R(t) = Average t-shirt cost t days from now
R(0) = $25
R'(0) = $5 per day
R' is decreasing, so R''(0) < 0
Daily Profit from Ethan's T-shirt shop is $5,000 and is increasing by $20 for each additional t-shirt (unit) sold. Use approximation by increments to estimate the profit when the number of units sold DECREASES by 20
LAF: F(x1) = F(x0) + F'(x0) * (Change in x)
F(x-20) = f(x0) + f'(x0)*(Change in x)
F(x-20) = $5,000 + $20(-20)
F(x-20) = 4600
Take the derivative of this equation
((x-2)/(2x+1))^9
5/(2x+1)^2
(45(x-2)^8)/(2x+1)^10
If they exist, find the critical values, intervals of increasing/decreasing, inflection values, and interval of concavity for the following function
f(x) = x^2-4x
Critical Value: x=2
Inc: (2,inf) Dec: (-inf, 2)
No inflection values. F''(x) is always positive so f(x) is always concave up (-inf,inf)
Ethan's math t-shirt store sells shirts has an initial average cost of $25. Due to the major increase in the love for mathematics, his average cost is increasing by $5 per day. But, because the midterm is coming up, the growth rate of his cost is decreasing.
Use Linear Approximation to estimate the the price per t-shirt after 2 weeks
LAF: F(x1) = F(x0) + F'(x0) * (Change in x)
R(14) = R(0) + R'(0) * 14
R(14) = 25 + 5 * 14
R(14) = 95
Remember these are approximates!
At Ethan's T-shirt company, x t-shirts are produced at a price of 1,050 - 5x
Estimate how much additional revenue is earned from increasing the number of units sold from 105 to 106 (Determine the marginal revenue)
R'(105)
R = p * x = (1,050 - 5x) x
R(x) = 1,050x - 5x2
R'(x) = 1,050 - 10x
R'(105) = 0