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Real-world applications

For the following exercises, explain the notation in words. The volume f ( t ) of a tank of gasoline, in gallons, t minutes after noon.

f ' ( 30 ) = −20

At 12:30 p.m. , the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.

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f ' ( 200 ) = 30

At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.

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For the following exercises, explain the functions in words. The height, s , of a projectile after t seconds is given by s ( t ) = 16 t 2 + 80 t .

s ( 2 ) = 96

The height of the projectile after 2 seconds is 96 feet.

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s ( 3 ) = 96

The height of the projectile at t = 3 seconds is 96 feet.

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s ( 0 ) = 0 , s ( 5 ) = 0.

The height of the projectile is zero at t = 0 and again at t = 5. In other words, the projectile starts on the ground and falls to earth again after 5 seconds.

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For the following exercises, the volume V of a sphere with respect to its radius r is given by V = 4 3 π r 3 .

Find the average rate of change of V as r changes from 1 cm to 2 cm.

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Find the instantaneous rate of change of V when r = 3  cm .

36 π

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For the following exercises, the revenue generated by selling x items is given by R ( x ) = 2 x 2 + 10 x .

Find the average change of the revenue function as x changes from x = 10 to x = 20.

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Find R ' ( 10 ) and interpret.

$50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.

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Find R ' ( 15 ) and interpret. Compare R ' ( 15 ) to R ' ( 10 ) , and explain the difference.

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For the following exercises, the cost of producing x cellphones is described by the function C ( x ) = x 2 4 x + 1000.

Find the average rate of change in the total cost as x changes from x = 10  to  x = 15.

$21 per unit

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Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16 th cellphone.

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Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21 st cellphone.

$36

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Extension

For the following exercises, use the definition for the derivative at a point x = a , lim x a f ( x ) f ( a ) x a , to find the derivative of the functions.

f ( x ) = 5 x 2 x + 4

f ' ( x ) = 10 a 1

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f ( x ) = x 2 + 4 x + 7

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f ( x ) = 4 3 x 2

4 ( 3 x ) 2

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Chapter review exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use [link] .

Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.

lim x −1 + f ( x )

2

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lim x −1 f ( x )

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lim x 1 f ( x )

does not exist

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At what values of x is the function discontinuous? What condition of continuity is violated?

Discontinuous at  x = 1 ( lim x a f ( x )  does not exist ) , x = 3   ( jump discontinuity ) , and  x = 7   ( lim x a f ( x )  does not exist ) .

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Using [link] , estimate lim x 0 f ( x ) .

x F ( x )
−0.1 2.875
−0.01 2.92
−0.001 2.998
0 Undefined
0.001 2.9987
0.01 2.865
0.1 2.78145
0.15 2.678

3

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For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a .   If the function has limit as x approaches a , state it. If not, discuss why there is no limit.

f ( x ) = { | x | 1 , i f x 1 x 3 , i f x = 1    a = 1

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f ( x ) = { 1 x + 1 , i f x = 2 ( x + 1 ) 2 , i f x 2    a = 2

lim x 2 f ( x ) = 1

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f ( x ) = { x + 3 , i f x < 1 x 3 , i f x > 1    a = 1

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Questions & Answers

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Practice Key Terms 7

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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