Solving Derivatives
Find g′(x) if g(x)=(3x2−x+8)3.
g'(x) = 3(3x2−x+8)2(6x−1)
The table below shows the value of a company’s stock price P(t) (in dollars) on the first of each month.
Month Jan Feb Mar
Price 42 45 50
Estimate the rate of change of the stock price at the start of February using the Average Rate of Change between January and March.
$4 per month
If C′(x)=12, interpret this in words.
The cost increases by $12 per unit produced.
If f′′(x) changes from positive to negative at x=2, what occurs?
There is an inflection point at x=2.
If C(q)=2q3−5q2+100, find the marginal cost function.
C′(q)=6q2−10q
Find f'(x) if f(x) = e2x ln(x).
f'(x) = e2x ln(x)・2 + e2x/x
The population of a town (in thousands) is given below.
Year 2022 2023 2024
Pop. 52 55 61
(a) Estimate the rate of change of the population during 2023.
(b) Use this rate to predict the population in 2025.
a) 4.5 thousand people per year
b) 65.5 thousand people
If s(t) represents the position of a car in meters and s′(t)=−12, what are the units of s′(t) and what does the negative sign mean?
Units: meters per second (m/s). The negative sign means the car is moving in the opposite direction (backward).
At x=5, f′(5)=0 and f′′(5)<0. What type of critical point is it?
a local maximum
Interpret R′(50)=310.
When 50 items are sold, revenue approximately increases by about $310 for the 51st item sold.
Find f′(x) if f(x)=2x+1 / x2+ex
f'(x) = (2)(x2+ex)−(2x+1)(2x+ex) / (x2+ex)2
Find the equation of the tangent line to f(x)=2x3+1 at x=2.
y=24(x-2)+17
You find R′(56)=478. Interpret in the context of hotel rooms.
When the hotel has rented 56 rooms, the approximate revenue from renting the 57th room will be $478.
Given f′(x)=6x(x−4), use the First Derivative Test to find where f(x) is increasing or decreasing, and identify any local extrema.
local max at x=0
local min at x=4
A company’s total cost function is C(q)=200+12q+0.5q2. Find C′(q) and interpret C′(40).
C'(40)=52
At 40 units, the cost of the 41st is approximately $52.
Find f′(x) if f(x)=√x+1 / x1/3
f′(x)=(x1/3)(1/2x−1/2)−(x1/2+1)(1/3x−2/3) / (x1/3)2
At x=1, f(1)=9 and f′(1)=−3. Use the tangent line to estimate f(1.5).
f(1.5) = 7.5
Given f(x)=x3−6x2+9x−2, determine if the function is increasing or decreasing and concave up or down at x=5.
Increasing and Concave Up
Given f′(x)=3x2−12x+9, use the Second Derivative Test to classify the critical points.
at x=1 local max
at x=3 local min
Given R(q)=450q+e0.14q, use R′(56) to estimate the revenue from booking the 57th room.
The estimate is $485.56
Find f′(x) if f(x)=ex^2+1ln(3x).
f′(x)=(ex^2+1・2x)ln(3x)+(ex^2+1)(1/3x・3)
If f(x)=e0.2x, find the tangent line at x=0.
y = 1 + 0.2x
If R′(q)=0 and R′′(q)<0, what does this tell you about revenue at that production level?
The revenue is at a local maximum
Given f′(x)=x3−6x2+9x, determine where f(x) is increasing/decreasing and find all local maxima and minima.
local min at x=0
Suppose revenue R(q)=400q+10e0.05q and cost C(q)=150q+2000.
(a) Find profit π(q)=R(q)−C(q).
(b) Find the marginal profit function π′(q).
(a) π(q)=250q+10e0.05q−2000
(b) π′(q)=250+0.5e0.05q