Basic Derivative Rules
Calculate the derivative:
f(x) = x3 + 4x2 - 6x + 7
f'(x) = 3x2 + 8x - 6
Calculate the derivative:
6 x3 - 4x
f(x) = -- + --------
x4 x1/2
f'(x) = -24x-5 + (5/2)x3/2 - 2x-1/2
At which x-value(s) does the graph of f(x) = x3 + 6x2 + 21x + 5 have a tangent line with slope equal to 12?
x = -1, -3
Find the critical numbers and classify them as local maximums or minimums:
f(x) = 2/3x3 - x2 - 24x - 4
x = 4, -3 are critical #s
local maximum when x = -3
local minimum when x = 4
Suppose if a certain item is sold at a price p = 3q - (2/q) (in dollars) then q is the number of items sold. Find the marginal revenue when q = 1.
R'(1) = 6 $/item
Calculate the derivative:
f(x) = (x3/2)(5x3-7)
f'(x) = 3/2x1/2 (5x3-7) + x3/2 (15x2)
Calculate the derivative:
f(x) = 4x ln(x2 + 2x)
(8x2 + 8x)
f'(x) = 4ln(x2 + 2x) + -----------
(x2 + 2x)
Write the equation of the line tangent to the graph of y = x2 - 3x + 3 at the point (2, 1)
y - 1 = 1(x-2) OR y = x -1
Find the critical numbers and intervals where f is increasing and decreasing:
f(x) = (2x + 4)1/3
Critical number at x = -2
Increasing on interval (-oo, oo)
Never decreasing
Suppose that the total cost (in hundreds of dollars) to produce x thousand barrels of a product is given by C(x) = 4x2 + 100x + 500. Find the marginal cost when x = 5. Describe what the marginal cost when x = 5 represents.
C'(5) = 140 hundred $ / thousand barrels
Means that when 5000 barrels are produced, total cost is increasing at a rate of $14,00 0 per thousand barrels produced.
Calculate the derivative:
(x3 + 0.5x)
f(x) = ------------
(5 - x7)
(5 - x7)(3x2 + 0.5) - (x3 + 0.5x)(-7x6)
f'(x) = ------------------------------------------
(5 - x7)2
Calculate the derivative:
ln(x)
f(x) = -------------
x4 + 3x -9
(x4 + 3x - 9) (1/x) - ln(x)(4x3+3)
f'(x) = -------------------------------------
(x4 + 3x - 9)2
Write the equation of the line tangent to the graph of y = 2ex + 2x at the point (0, 2)
y-2 = 4(x - 0) OR y=4x+2
Find the absolute extrema for f(x) = x4 - 18x2 + 1 on [-1, 4]. Explain why we already know, before doing any work, that f has an absolute minimum and maximum on [-1, 4].
Absolute maximum is 1 when x = 0
Absolute minimum is -80 when x = 3
Suppose that the total cost C(x) to manufacture a quantity of x thousand gallons of perfume per week is given by C(x) = x4 - 4/3x3 - 4x2 + 1. For what values of x is the cost C(x) increasing? How many gallons of perfume should they make to minimize cost?
Increasing on interval (2, oo)
To minimize cost, 2000 gallons should be produced to minimize cost
Calculate the derivative:
f(x) = (4 + 21/2x)5
f'(x) = 5(4 + 21/2x)4 (21/2)
Calculate the derivative:
f(x) = (5 - x2)3/2 - ln(2 - 5x3) - e
15x2
f'(x) = -3x (5 - x2)1/2 + -------- 2 - 5x3
Use the limit definition of the derivative to find f′(x) for f(x) = 3x + x2
f'(x) = 3 + 2x
(Ava must verify you used limit notation)
Find the x-value of all points where the function f(x) = x2 - 16/x has absolute extrema on the interval [1, 4].
Absolute maximum when x = 4
Absolute minimum when x = 1
If the price charged for an item is p(x) = 12 - x/8 dollars, then x thousand items will be sold. Find (a) the total revenue function R(x), (b) the value of x that corresponds to maximal revenue, (c) the price that should be charged per item to maximize revenue
a) R(x) = 12000x - 125x2
b) x = 48
c) $6.00
Calculate the derivative:
f(x) = e3x^4 + x - 2
f'(x) = e3x^4 + x - 2 (12x3 + 1)
Calculate the derivative:
f(x) = e3x+1 (7-x6)4
f'(x) = 3e3x+1 (7 - x6)4 - 24x5 e3x+1 (7 - x6)3
Put each term into one of the following groups: Interpretations of f'(x), interpretations of y2 - y1 / x2 - x1 and reasons a function might be discontinuous.
Jump, derivative, instantaneous rate of change, asymptote, slope of the secant line, slope of the tangent line, hole, average rate of change
f'(x): derivative, instantaneous rate of change, slope of tangent
y2-y1 / x2-x1: slope of secant, average rate of change
discontinuous: jump, asymptote, hole
For the function f(x) = x4 - 4x3, determine the intervals on which the graph of f is concave up and down and find the points of inflection for the curve.
Concave up on (-oo, 0) U (2, oo)
Concave down on (0, 2)
Inflection points: (0,0) and (2, -16)
(Worth 700 Points)
The cost and demand functions for q pizzas per day are C(q) = 20 + 4q, p = D(q) = 10 - 1/2q.
(a) Find the total revenue and total profit functions, as well as the marginal profit function.
(b) Calculate the number of pizzas demanded that maximizes profit.
(c) What price should I set my pizzas in order to meet that demand?
(d) What is the maximum possible profit?
(a) R(q) = 10q - 1/2q2, P(q) = 1/2q2 + 6q - 20, P'(q) = -q + 6
(b) = 6 pizzas
(c) = $7
(d) = $-2