Derivative Rules
What derivative rule is this? Solve f'(x) when f(x) = 2x4+34x3-2x-2+2x-4
Power Rule
8x3+ 102x2 + 4x-3+ 2
What is the profit function?
P(x)=R(x)-C(x)
Factor 45 + 6x - 3x2
Rewrite; -3x2+ 6x + 45
-3(x2 - 2x - 15)
-3(x + 2)(x - 5)
What are critical values?
Where f'(x) = 0 or undefined
Where the max/min is
Where you are at top of hill, bottom of valley.
What derivative rule is this? Solve for f'(x) when f(x) = (4x3+ 9x2+ 3)4
Chain Rule
4(4x3+ 9x2+ 3)3 * (12x2 + 18x)
What is P'(x)?
Marginal Profit
Find the critical values for f(x) = 45 + 6x - 3x2
f'(x) = -3(-15 - 2x - x2)
-3(x + 2)(x - 5)
x = -2, x = 5
In a typical optimization problem (max/min problem), we want to find a relative maximum or relative minimum of a function. Our process is to...
1. find the derivative of a function
2. set the derivative equal to 0
3. solve for x
What derivative rule is this? Solve for f'(x) if f(x) = (t2 + 42t - t)(t3 - 2t + t)
Product Rule
(t2 + 42t - t)*(3t2 -2 + 1) + (t3 - 2t + t)*(2t + 42 - 1)
What does R(x) mean, and what does R'(x) represent?
R(x) is the total revenue, where x = (# of units sold) and p = (price per unit) when multiplied
R(x) = x * p
R'(x) represents the prediction of the revenue of the next additional unit sold where
R'(x) = Marginal Revenue and/or Derivative of R(x)
What does MP, MR, and MC represent?
Marginal Profit, Marginal Revenue, Marginal Cost
Find the critical value(s) for the function:
y = (x3/3) + (x2/2) - 2x +4
Rewrite = (1/3x3)+ (1/2x2)- 2x + 4
y'= x2 + x - 2
x2 + x - 2 = 0
(x - 1)(x + 2) = 0
x = 1, x = -2
What derivative rule is this? Solve for y' if y = 4x5- √x + (5 /√ x) + (1/x6)
Power Rule
20x4 - (1/2x-1/2) + (5/2x-1/2) - (1x-7)
What does C(x) mean, and what does C'(x) represent?
C(x) is the total cost function of production
C'(x) = Marginal Cost and/or Derivative of C(x)
What derivative rule is this? If f(x) = 1/2(x3+4)2 Find f'(x).
Rewrite: 1/2 (x3+4)-2
-1(x3+4)-3*(3x2)
A firm has a total revenue given by R(x) = 2800x - 8x2 - x3 for x units of a product. If only 40 units are sold per day, find the revenue, and the marginal revenue. Should the company increase their profits based on the information?
R'(40) = 2800 - 16(40) - 3(40)2 = $-2640/unit
R(40) = 2800(40) - 8(40)2-(40)3= $35200
No because although they are making a profit of $35,200 at 40 units, in the long run or the next additional unit that they produce they will lose revenue of $2640.
Find the critical values, and the relative maximum of f(x)= x3- 3x + 3
f'(x) = 3x2- 3 =0. f(x)= (1)3- 3(1) + 3 = 1
3x2 = 3
√x2 = √3
x = +/- (1)
What derivative rule is this? f(x) = (24x3 + 12x2 +6x - 3)7 Find f'(x).
Chain Rule
7(24x3 + 12x2 +6x - 3)6 * (72x2 + 24x + 6)
A company sells x units of a product each month.
The price per unit is p = 70 + 0.1x dollars per unit
Find R(x), Find R'(x) and interpret what they both mean when x = 30
R(x)= 70x + 0.1x2
R'(x) = 70 + 0.2x
R(30) = 70(30) + 0.1(30)2 = 2190 (Total revenue from 30 units sold and produced)
R'(30) = 70 + 0.2(30) = 76 (Where the total is the prediction of revenue of the 30th unit)
What derivative rule is this? f(x) = (12x2 - 43x)/(x2- 3x). Find f'(x).
Quotient Rule; (x2 - 3x)(24x - 43) - (12x2 - 43x)(2x - 3)/ (x2 - 3x)2
LoDiHi-HiDLo, draw the line, and square BLo (Below)
The monthly demand function for x units of a product sold by a monopoly is p= 8000 - x, the average cost is C(x) = 5x + 4000. Find the profit function, and the selling price at this optimal quantity. What is the maximum profit?
Where C(x) = (5x+4000)(x)
and R(x) = p * x
Hint: Practice Problem #37 10.3
P(x) = 8000x - x2 - 5x2 - 4000x
Profit Function; P(x) = 4000x - 6x2
P'(x) = 4000 - 12x = 0, -4000/12 = x, where x = $333.33 per unit (critical value) where at x = 333.33 will maximize profit.
Max Profit; P(333.33) = 4000(333.33) - 6(333.33)2 = $666666.67
Selling price; p = 8000 - 333.33 = $7666.67 per unit.
Find the derivative, critical values and/or relative maximum and minimum for y = 1/2x2 - x
y' = x -1
C.V. = x =1
y(1)= 1/2(1)2 - 1 = -1/2
Rel Minimum (1, -1/2)