Chapter 0
Functions and Graphs
0.2
Functions and Models
Exercise 1. A function f is given by f (x) = 3x2 − 2x + 8. Find f (0), f (−5)
and f (7a).
Exercise 2. Recall the difference quotient,
D.Q. =
f (x + h) − f (x)
.
h
Given f (x) = x − x2 , find f (4), f (x + h) and the D.Q.
Exercise 3. Graph the function defined


4
f (x) = 3 − x2


2x − 6
0.3
as:
for x ≤ 0
for 0 < x ≤ 2
for x > 2
Finding Domain and Range
Recall Set Notation and Interval Notation.
Exercise 4. Write interval notation for each set of graph:
a. {x| − 4 < x < 5}
b. {x|x ≥ −2}
c. {x| − 2 < x ≤ 4} draw as a picture.
d. {x|x < −1} draw as a picture.
Exercise 5. Find the domain and range of the following functions:
1
a. f (x) is a polynomial
b. f (x) = |x|
c. f (x) =
d. f (x) =
0.4
3
2x−5
√
4 + 3x
Slope and Linear Functions
Recall the most basic lines, horizontal and vertical lines, are given by the formulas: y = c and x = c respectively where c ∈ R. Clearly all vertical lines are
not functions.
Definition 0.1. The variable y varies directly as x if there is some constant
m such that y = mx. We also say that y is directly proportional to x.
Exercise 6. The weight M , in pounds, of an object on the moon is directly
proportional to the weight E of that object on Earth. An astronaut who weighs
180 lb on Earth will weigh 28.8 lb on the moon.
a. Find an equation of variation
b. An astronaut weighs 120 lb on Earth. How much will he weigh on the moon?
Definition 0.2. The slope-intercept form of a line is given by y = mx + b
where m is the slope and b is the y-intercept.
Definition 0.3. The point-slope form of a line is given by y − y1 = m(x − x1 )
where m is the slope and (x1 , y1 ) is a point on the line.
Definition 0.4. The slope, m, of a line going through the points (x1 , y1 ) and
1
(x2 , y2 ) is given by m = xy22 −y
−x1 .
Exercise 7. Find the equation of the line given the following data:
a. The line has slope 3 and goes through the point (−2, 6).
b. The line goes through the points (−2, −7) and (3, 3).
Recall the relation between cost, revenue and profit. P (x) = R(x) − C(x).
0.5
Nonlinear Functions and Models
Definition 0.5. A quadratic function f is given by
f (x) = ax2 + bx + c
where a 6= 0.
The graph of a quadratic function f (x) = ax2 + bx + c is called a parabola.
2
1. It is always a cup-shaped curve.
2. It opens upward if a > 0 or opens downward if a < 0.
b
b
3. It has a turning point, or vertex, whose coordinates are given by − 2a
, f − 2a
b
is the line of symmetry.
4. The line x = − 2a
Theorem 0.1. The solutions of any quadratic equations ax2 +bx+c = 0, a 6= 0,
are given by
√
−b ± b2 − 4ac
x=
.
2a
Exercise 8. Graph f (x) = x2 − 2x − 3.
Exercise 9. Solve 3x2 + 4x = 2.
Mention iPhone apps and stuff for graphing.
Definition 0.6. A polynomial function f is given by
f (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 ,
where n is a nonnegative integer and ai is a real number for all 0 ≤ i ≤ n, called
a coefficient.
Definition 0.7. Functions given by the quotient of polynomials are called rational functions.
Exercise 10. Graph
a. f (x) =
x2 −9
x+3
b. f (x) =
2
x−3
Definition 0.8. y varies inversely as x if there is some number k such that
y = xk . We also say that y is inversely proportional to x.
Exercise 11. Simplify:
a. 85/3
b. 813/4
Exercise 12. Find the equilibrium point for the demand and supply functions
for the Ultra-Fine coffee maker. Here q represents the number of coffee makers
produced, in hundreds, and x is the price, in dollars.
Demand: q = 50 − 14 x.
Supply: q = x − 25
3
.
0.6
Mathematical Modeling and Curve Fitting
Exercise 13. The following table shows the number of U.S. households with
cable television and a scatterplot of the data. It appears that the data can be
represented or modeled by a linear function.
Pick the points x = 2 and x = 6.
Year, x
1999, 0
2000, 1
2001, 2
2002, 3
2003, 4
2004, 5
2005, 6
2006, 7
Number of TV subscribers (in millions), y
76.4
78.6
81.5
87.8
88.4
92.4
94.0
95.0
4
Chapter 1
Differentiation
1.1
Limits: A Numerical and Graphical Approach
Definition 1.1. As x approaches a, the limit of f (x) is L, written lim f (x) = L,
x→a
if all values of f (x) are close to L for values of x that are sufficiently close, but
not equal, to a. The limit L must be a unique real number
Make note of left and right hand limits
Theorem 1.1.
⇐⇒
lim f (x) = lim− = L
x→a+
x→a
lim f (x) = L
x→a
Exercise 1. Find the limit, lim f (x) using the graphs of the following functions.
x→1
f (x) = x + 2
(
3x − 2,
f (x) =
x + 3,
x2 − 1
x−1
(
5,
f (x) =
x + 1,
x≤1
x>1
f (x) =
5
x=1
x 6= 1
Exercise 2. Given f (x) = x1 , find the following:
a. lim+ f (x)
x→0
b. lim− f (x)
x→0
c. lim f (x)
x→0
d. lim f (x)
x→∞
e.
lim f (x)
x→−∞
1.2
Algebraic Limits and Continuity
Theorem 1.2. For any rational function F , with a in the domain of F ,
lim F (x) = F (a).
x→a
Theorem 1.3.
L1 The limit of a constant is the constant.
L2 The limit of a power is the power of the limit, and the limit of a root is the
root of that limit.
L3 The limit of a sum or difference is the sum or difference of the limits.
L4 The limit of a product is the product of the limits.
L5 The limit of a quotient is the quotient of the limits.
Exercise 3. Find the following limits:
a. lim 2x2 − 5x + 7
x→2
x2 −1
x→1 x−1
b. lim
c. lim
x→0
√
x2 − 3x + 2
Definition 1.2. A function f is continuous at x = a if and only if:
a. f (a) exists
b. lim f (x) exists
x→a
c. lim f (x) = f (a).
x→a
A function is continuous over an interval I if it is continuous at each point
of I.
6
Exercise 4. Determine if the following functions are continuous at the given
points.
a. f (x) = 2x + 3 at x = 4
b.
(
f (x) =
1
2x
+ 3,
x − 1,
x < −2
x ≥ −2
at x = −2?
c.
(
−x + 3, x ≤ 2
f (x) =
x2 − 3
x>2
at x = 2?
1.3
Average Rates of Change
Definition 1.3. The average rate of change of y with respect to x, as
x changes from x1 to x2 , is the ratio of the change in output to the change in
input:
y2 − y1
where x1 6= x2 .
x2 − x1
Exercise 5. For y = f (x) = x2 , find the average rate of change as:
a. x = 1 to 3
b. x = 1 to 2
c. x = 2 to 3
Definition 1.4. The average rate of change of f with respect to x is also called
the difference quotient. It is given by
f (x + h) − f (x)
,
h
where h 6= 0
Exercise 6. For f (x) = x2 , find the difference quotient when:
a. x = 5 and h = 3.
b. x = 5 and h = .1.
c. x = 5 and h = .01.
Exercise 7. Find and simplify the difference quotient for:
a. f (x) = x3
b. f (x) =
c. f (x) =
1
x
√
x
7
1.4
Differentiation Using Limits of Difference Quotients
Definition 1.5. For a function y = f (x), its derivative at x is the function
f 0 defined by
f (x + h) − f (x)
f 0 (x) = lim
,
h→0
h
provided the limit exists. If f 0 (x) exists, then we say that f is differentiable
at x. We sometimes call f 0 the derived function.
Note: f 0 (x) is read “derivative of f at x,” “f prime of x,” or “f prime at x.”
Exercise 8. For the given functions, find f 0 (x). Then find f 0 (−3) and f 0 (4).
a. f (x) = x2
b. f (x) = x3
c. f (x) = 3x − 4
d. f (x) =
1
x
Exercise 9. Example of a non-differentiable function. Consider f (x) = |x|.
1.5
Differentiation Techniques: The Power and
Sum-Difference Rules
Definition 1.6. Let y = f (x) be a function of x. Then the following all repdy
d
resent the derivative of y with respect to x: y 0 = f 0 (x) = dx
= dx
f (x). The
fractional notation is known as Leibniz notation. The other is known as ‘prime’
notation.
Theorem 1.4 (Power Rule). For any real number k, if y = xk , then
d k
x = k · xk−1 .
dx
Exercise 10. Differentiate the following using the power rule.
a. y = x5
b. y = x
c. y = x−4
√
d. y = 5 x
e. y = x0.7
8
Theorem 1.5. The derivative of a constant function is 0. That is,
d
c = 0.
dx
Theorem 1.6. The derivative of a constant times a function is the constant
times the derivative of the function. Using derivative notation, we can write
this as
d
d
[c · f (x)] = c ·
f (x)
dx
dx
Theorem 1.7. The derivative of a sum or difference is the sum or difference
of the derivatives.
d
d
d
[f (x) ± g(x)] =
f (x) ±
g(x).
dx
dx
dx
Exercise 11. Find the derivative of the following functions:
a. 5x3 − 7
√
b. 24x − x +
5
x
Exercise 12. Find when the slope of the tangent line is horizontal to f (x) =
−x3 + 6x2 .
Elaborate on increasing and decreasing functions and the relation to the derivative. Also, find the equation of a line tangent...
1.6
Differentiation Techniques: The Product and
Quotient Rules
Theorem 1.8. Let F (x) = f (x) · g(x). Then
d
[f (x) · g(x)] = f (x) · g 0 (x) + g(x) · f 0 (x).
dx
Exercise 13. Find the derivative of the following functions:
F 0 (x) =
a. (x4 − 2x3 − 7)(3x2 − 5x)
√
b. (x2 + 4x − 11)(7x3 − x)
Theorem 1.9. If Q(x) =
N (x)
D(x) ,
Q0 (x) =
then
D(x) · Q0 (x) − N (x) · D0 (x)
.
[D(x)]2
Exercise 14. Find the derivative of the following functions:
a. f (x) =
b.
x5
x2
x2 −3x
x−1
Definition 1.7. If C(x) is the cost of producing x items, then the average
cost of producing x items is C(x)
x . Average revenue and average profit are
defined similarly.
9
1.7
The Chain Rule
Theorem 1.10 (Extended Power Rule). Suppose that g(x) is a differentiable
function of x. Then, for any real number k,
d
d
[g(x)]k = k[g(x)]k−1 ·
g(x).
dx
dx
Exercise 15. Differentiate:
a. f (x) = (1 + x3 )1/2
b. f (x) = (3x − 5)4 (7 − x)10
q
x+3
c. f (x) = 4 x−2
Definition 1.8. The composed function f ◦ g, the composition of f and g,
is defined as
(f ◦ g)(x) = f (g(x)).
Theorem 1.11. The derivative of the composition f ◦ g is given by
d
d
[(f ◦ g)(x)] =
[f (g(x))] = f 0 (g(x)) · g 0 (x).
dx
dx
1.8
Higher-Order Derivatives
We can express finding the derivative of the derivative with the following notation:
h i
dy
d2 y
d
The second derivative of y = f (x) as f 00 (x) = dx
dx = dx2 .
h 2 i
d y
d3 y
d
The third derivative of y = f (x) as f 000 (x) = dx
dx2 = dx3 .
etc...
Exercise 16. Find
a. f (x) =
d2 y
dx2
for
1
x
b. y = (x2 + 10x)20
Definition 1.9. Acceleration = a(t) = v 0 (t) = s00 (t).
Exercise 17. For s(t) = 10t2 , find v(t) and a(t), where s is the distance from
the starting point, in miles, and t is in hours. Then find the distance, velocity,
and acceleration at t = 4 hours.
10
2
Exercise 18. Given N (t) = 250,000t
(2t+1)2 , t > 0, represents the quantity sold N of
a product after t weeks on the market. Its derivative is
N 0 (t) =
500, 000t
.
(2t + 1)3
Recall that N 0 (t) represents the rate of change in number of units sold per week.
Find N 00 (t); then use it to calculate N 00 (52) and N 00 (208) and interpret these
results.
11
Chapter 2
Applications of
Differentiation
2.1
Using First Derivatives to Find Maximum
and Minimum Values and Sketch Graphs
Definition 2.1. A function f is increasing over I if, for every a and b in I,
if a < b, then f (a) < f (b).
A function f is decreasing over I if, for every a and b in I, if a < b, then
f (a) > f (b).
Theorem 2.1. If f 0 (x) > 0 for all x in an open interval I, then f is increasing
over I.
If f 0 (x) < 0 for all x in an open interval I, then f is decreasing over I.
Definition 2.2. A critical value of a function f is any number c in the
domain of f for which the tangent line at (c, f (c)) is horizontal or for which
the derivative does not exist. That is, c is a critical value if f (c) exists and
f 0 (c) = 0 or does not exist.
Definition 2.3. Let I be the domain of f .
f (c) is a relative minimum if there exists within I an open interval I1 containing c such that f (c) ≤ f (x), for all x in I1 ;
and
f (c) is a relative maximum if there exists within I an open interval I2 containing c such that f (c) ≥ f (x), for all x in I2 .
Theorem 2.2. If a function f has a relative extreme value f (c) on an open
interval, then c is a critical value; that is f 0 (c) = 0 or does not exist.
12
Theorem 2.3 (First Derivative Test). For any continuous function f that has
exactly one critical value c in an open interval (a, b):
F1. f has a relative minimum at c if f 0 (x) < 0 on (a, c) and f 0 (x) > 0 on (c, b).
i.e. f is decreasing to the left of c and increasing to the right of c.
F2. f has a relative maximum at c if f 0 (x) > 0 on (a, c) and f 0 (x) < 0 on (c, b).
i.e. f is increasing to the left of c and decreasing to the right of c.
F3. f has neither a relative maximum nor a relative minimum at c if f 0 (x) has
the same sign on (a, c) as on (c, b).
Exercise 1. Graph the given function and find all relative extrema.
a. f (x) = 2x3 − 3x2 − 12x + 12
b. f (x) = 2x3 − 3x4
c. f (x) = (x − 2)2/3 + 1
2.2
Using Second Derivatives to Find Maximum
and Minimum Values and Sketch Graphs
Definition 2.4. Suppose that f is a function whose derivative f 0 exists at every
point in an open interval I. Then
f is concave up on I if f 0 is increasing over I,
f is concave down on I if f 0 is decreasing over I.
Theorem 2.4.
1. If f 00 (x) > 0 on an interval I, then the graph of f is concave up.
2. If f 00 (x) < 0 on an interval I, then the graph of f is concave down.
Theorem 2.5 (Second Derivative Test). Suppose that f is differentiable for
every x in an open interval (a, b) and that there is a critical value c in (a, b) for
which f 0 (c) = 0. Then:
1. f (c) is a relative minimum if f 00 (c) > 0.
2. f (c) is a relative maximum if f 00 (c) < 0.
For f 00 (c) = 0, the First Derivative Test can be used to determine whether f (x)
is a relative extremum.
Show graphs of f (x) = (x − 2)4 + 1 and f (x) = (x − 2)3 + 1 as examples for
the f 00 (c) = 0 case.
Exercise 2. Find the relative extrema and sketch a graph of the function f
given by:
13
a. f (x) = x3 + 3x2 − 9x − 13
b. f (x) = 3x5 − 20x3
Definition 2.5. A point of inflection is a point across which the direction
of concavity changes.
Theorem 2.6. If a function f has a point of inflection, it must occur at a point
x0 where f 00 (x0 ) = 0 or does not exist.
Exercise 3. Given f (x) = (2x − 5)1/3 + 1, list the coordinates of any and
all extrema and points of inflection. State where the function is increasing or
decreasing, as well as where it is concave up or concave down.
2.3
Graph Sketching: Asymptotes and Rational
Functions
Definition 2.6. A rational function is a function f that can be described by
f (x) =
P (x)
Q(x)
where P (x) and Q(x) are polynomials, with Q(x) no the zero polynomial. The
domain consists of all inputs x for which Q(x) 6= 0.
Definition 2.7. The line x = a is a vertical asymptote if any of the following
limit statements is true
lim f (x) = ±∞
or
x→a−
lim f (x) = ±∞
x→a+
The graph of a rational function never crosses a vertical asymptote. If the
expression that defines the rational function f is simplified, meaning that it has
no common factor other than −1 or 1, then if a is an input that makes the
denominator 0, the line x = a is a vertical asymptote.
Exercise 4. Determine the vertical asymptotes:
a. f (x) =
3x−2
x(x−5)(x+3)
b. f (x) =
x2 −2x
x3 −x
Definition 2.8. The line y = b is a horizontal asymptote if either or both
of the following limit statements is true:
lim f (x) = b.
x→±∞
The graph of a rational function may or may not cross a horizontal asymptote.
Horizontal asymptotes occur when the degree of the numerator is less than or
equal to the degree of the denominator.
14
Exercise 5. Determine the horizontal asymptotes of the given functions:
a. f (x) =
3x−4
x
b. f (x) =
3x2 +2x−4
2x2 −x+1
c. f (x) =
2x−3
x3 −3x2 +4
Definition 2.9. A linear asymptote that is neither vertical nor horizontal is
called a slant asymptote, or an oblique asymptote. For any rational function of the form f (x) = p(x)
q(x) , a slant asymptote occurs when the degree of p(x)
is exactly 1 more than the degree of q(x). A graph can cross a slant asymptote.
Exercise 6. Find the slant asymptote:
f (x) =
x2 − 4
x−1
Exercise 7. Sketch the graph of
a. f (x) =
8
x2 −4
b. f (x) =
x2 +4
x
2.4
Using Derivatives to Find Absolute Maximum and Minimum Values
Definition 2.10. Suppose that f is a function with domain I.
f (c) is an absolute minimum if f (c) ≤ f (x) for all x in I.
f (c) is an absolute maximum if f (c) ≥ f (x) for all x in I.
Theorem 2.7 (Extreme Value Theorem). A continuous function f defined over
a closed interval [a, b] must have an absolute maximum value and an absolute
minimum value over [a, b].
Theorem 2.8 (Max-Min Principle). Suppose that f is a continuous function
defined over a closed interval [a, b]. To find the absolute maximum and minimum
values over [a, b]:
1. First find f 0 (x).
2. Then determine all critical values in [a, b].
3. List the values from step 2. and the endpoints of the interval: a, c1 , c2 , . . . , cn , b.
4. Evaluate f (x) for each value in step 3.: f (a), f (c1 ), f (c2 ), . . . , f (cn ), f (b).
The largest of these is the absolute maximum of f over [a,b]. The
smallest of these is the absolute minimum of f over [a,b].
15
Exercise 8. Find the absolute maximum and minimum for the function over
the given closed interval:
a. f (x) = x3 − 3x + 2 over [−2, 1.5]
b. f (x) = x3 − 3x + 2 over [−3, −1.5]
Theorem 2.9 (Max-Min Principle 2). Suppose that f is a function such that
f 0 (x) exists for every x in an interval I and that there is exactly one (critical)
value c in I, for which f 0 (c) = 0. Then
f (c) is the absolute maximum value over I if f 00 (c) < 0
or
f (c) is the absolute minimum value over I if f 00 (c) > 0.
Exercise 9. Find the absolute maximum and minimum values for the function,
if they exist, over the given interval:
a. f (x) = 3x − x2
b. f (x) = (x − 2)3 + 1
c. f (x) = 5x +
2.5
35
x
for (0, ∞).
Maximum-Minimum Problems; Business and
Economics Applications
Exercise 10. A manufacturer of food-storage containers makes a cylindrical can
with a volume of 500 mL. What dimensions (height and radius) will minimize
the material needed to produce each can; that is, minimize the surface area?
(Recall 1 mL = 1 cm3 ). Answer: r ≈ 4.3 cm.
Exercise 11. A stereo manufacturer determines that in order to sell x units of
a new stereo, the price per unit, in dollars, must be
p(x) = 1000 − x.
The manufacturer also determines that the total cost of producing x units is
given by
C(x) = 3000 + 20x
a. Find the total revenue R(x).
b. Find the total profit P (x).
c. How many units must the company produce and sell in order to maximize
profit?
d. What price per unit must be charged in order to make this maximum profit?
16
Theorem 2.10. Maximum profit occurs at those x-values for which
R0 (x) = C 0 (x)
and
R00 (x) < C 00 (x)
Exercise 12. Promoters of international fund-raising concerts must walk a fine
line between profit and loss, especially when determining the price to charge for
admission to closed circuit TV showings in locals theaters. By keeping records,
a theater determines that at an admission price of $26, it averages 1000 people
in attendance. For every drop in price of $1, it gains 50 customers. Each
customer spends an average of $4 on concessions. What admission price should
the theater charge in order to maximize total revenue?
2.7
Implicit Differentiation and Related Rates
Consider the equation y 3 = x. √
This equation implies that y is a function of x,
as we can clearly see that y = 3 x = x1/3 . We also can easily show that
dy
1
= x−2/3 .
dx
3
dy
A method know as implicit differentiation allows us to find dx
without solving for y. To do so, we use the chain rule, treating y as a function of x and
differentiating both sides of y 3 = x with respect to x:
d 3
d
y =
x
dx
dx
dy
=1
3y 2
dx
Solving for
dy
dx ,
we get
1
1
1
1
dy
= 2 = y −2 = (x1/3 )−2 = x−2/3
dx
3y
3
3
3
It is often difficult or impossible to solve for y explicitly or even express
in terms of x.
dy
dx
solely
Exercise 13. For y 3 + x2 y 5 − x4 = 27:
a. Find
dy
dx
using implicit differentiation.
b. Find the slope of the tangent line to the curve at the point (0, 3).
p
Exercise 14. For the demand equation x = 200 − p3 , differentiate implicitly
dp
to find dx
.
Exercise 15. A restaurant supplier services the restaurants in a circular area
in such a way that the radius r is increasing at the rate of 2 miles per year at
the moment when r = 5 miles. At that moment, how fast is the area increasing?
17
Chapter 3
Exponential and
Logarithmic Functions
3.1
Exponential Functions
Definition 3.1. An exponential function f is given by
f (x) = ax ,
where x is any real number, a > 0 and a 6= 1. The number a is called the base.
Try to find the derivative of f (x) = ax . Look at the values specifically for
a = 2 and a = 3. Then elaborate to find e.
Definition 3.2.
e = lim (1 + h)1/h ≈ 2.718281828459
h→0
We call e the natural number. It is also known as Euler’s constant.
Theorem 3.1. The derivative of the function f given by f (x) = ex is itself:
f 0 (x) = f (x),
or
d x
e = ex .
dx
Exercise 1. Find the derivative of the following functions:
a. y = 3ex
b. y = x2 ex
c. y =
ex
x3
Theorem 3.2. The derivative of e to some power is the product of e to that
power and the derivative of the power.
d f (x)
df
e
= ef (x) · f 0 (x) = ef (x) ·
dx
dx
18
Exercise 2. Differentiate each of the following with respect to x:
a. y = e8x
b. y = e−x
√
c. e
2
+4x−7
x2 −3
Exercise 3. Graph g(x) = e−x . Analyze the graph using calculus.
3.2
Logarithmic Functions
Definition 3.3. A logarithm is defined as follows:
loga x = y
means ay = x, a > 0, a 6= 1.
The number loga x is the power y to which we raise a to get x. The number a
is called the logarithmic base.
Theorem 3.3 (Properties of Logarithms). For any positive numbers M , N , a,
and b, with a, b 6= 1, and any real number k:
P1. loga (M N ) = loga M + loga N
P2. loga
M
N
= loga M − loga N
P3. loga (M k ) = k · loga M
P4. loga a = 1
P5. loga ak = k
P6. loga 1 = 0
P7. logb M =
loga M
loga b
Exercise 4. Given loga 2 = 0.301 and loga 3 = 0.477, find the following:
a. loga 6
b. loga 81
√
c. loga a
d. loga
1
3
Definition 3.4. for any positive number x, log x = log10 x. This is known as
the common logarithm.
Definition 3.5. For any positive number xm ln x = loge x. This is know as the
natural logarithm.
19
Exercise 5. Solve for t:
a. et = 40
b. e−0.04t = 0.05
Theorem 3.4. For any positive number x,
d
1
ln x = .
dx
x
Exercise 6. Find the derivative of y = x2 ln x + 5x.
Theorem 3.5.
1
d
f 0 (x)
ln f (x) =
· f 0 (x) =
dx
f (x)
f (x)
Exercise 7. Differentiate:
a. y = ln(3x)
b. y = ln(x2 − 5)
3 c. y = ln x x+4
3.3
Applications: Uninhibited Growth and Limited Growth Models
Theorem 3.6. A function y = f (x) satisfies the equation
dy
= ky or f 0 (x) = k · f (x)
dx
if and only if
y = cekx or f (x) = cekx
for some constant c.
Exercise 8. Find the general form of the function that satisfies the equation
dA
= 5A.
dt
Exercise 9. Find the general form of the function that satisfies the equation
f 0 (z) = 8 · f (z).
20
Exercise 10. Suppose that an amount P0 , in dollars, is invested in the Von
Neumann Hi-Yield Fund, with interest compounded continuously at 7% per
year. That is, the balance P grows at the rate given by
dP
= 0.07P.
dt
a. Find the function that satisfies the equation. Write it in terms of P0 and
0.07.
b. Suppose that $100 is invested. What is the balance after 1 year?
c. In what period of time will an investment of $100 double itself?
Theorem 3.7. The growth rate k and the doubling time T are related by
ln 2
ln 2
or
T =
T
k
Exercise 11. Membership in Facebook has been doubling every 6 months.
What is the exponential growth rate of Facebook membership, as a percentage?
kT = ln 2
or
k=
Mention the “Rule of 70” and how it’s derived using this formula.
Exercise 12. A 1939 comic book with the first appearance of the “Caped
Crusader,” Batman sold at auction in Dallas in 2010 for a record $1.075 million. The comic book originally cost 10¢. Using two representative data points
(0, $0.10) and (71, $1, 075, 000), we can find an exponential function that models
the increasing value of the comic book. The modeling assumption is that the
value V of the comic book has grown exponentially, as given by
dV
= kV.
dt
(Source: Heritage Auction Galleries.)
a. Find the function that satisfies this equation. Assume that V0 = $0.10.
b. Estimate the value of the comic book in 2020.
c. What is the doubling time for the value of the comic book?
d. In what year will the value of the comic book be 30 million, assuming there
is no change in the growth rate?
The growth model P (t) = P0 ekt has many applications to unlimited population growth. However, there are often factors that prevent a population from
exceeding some limiting value L– perhaps a limitation on food, living space, or
other natural resources. A model for such growth is the logistic equation or
logistic function which is given by
L
, for k > 0.
1 + be−kt
Another model of limited growth is provided by the function
P (t) =
P (t) = L(1 − e−kt ), for k > 0.
21
Exercise 13. In a town whose population is 3500, an epidemic of a disease
occurs. The number of people N infected t days after the disease first appears
is given by
3500
.
N (t) =
1 + 19.9e−0.6t
a. How many people are initially infected with the disease?
b. Find the number infected after 2 days, 5 days, and 18 days.
c. Graph the equation.
d. Find the rate at which the disease is spreading after 16 days.
3.4
Applications: Decay
Exercise 14. Strontium-90 has a decay rate of 2.8% per year. The rate of
change of an amount N of this radioactive isotope is given by
dN
= −0.028N.
dt
a. Find the function that satisfies the equation. Let N0 represent the amount
present at t = 0.
b. Suppose that 1000 grams (g) of strontium-90 is present at t = 0. How much
will remain after 70 year?
c. After how long will half of the 1000 g remain?
Theorem 3.8. The decay rate, k, and the half-life, T , are related by
kT = ln 2 or k =
ln 2
ln 2
or T =
.
T
k
Exercise 15. Plutonium-239, a common product of a functioning nuclear reactor, can be deadly to people exposed to it. Its decay rate is about 0.0028%
per year. What is its half-life?
Theorem 3.9. The present value P0 of an amount P due t years later, at
interest rate k, compounded continuously, is given by
P = P0 e−kt .
Theorem 3.10. The temperature T of a cooling object drops at a rate that is
proportional to the difference T − C, where C is the constant temperature of the
surrounding medium. Thus,
dT
= −k(T − C).
dt
The function that satisfies this equation is
T = T (t) = ae−kt + C.
22
Exercise 16. Found stabbed by a collection of number 2 pencils, Prof. Cal
Kulice’s body was slumped over a stack of exams with plenty of red marks on
them. A coroner arrives at noon, immediately takes the temperature of the
body, and finds it to be 94.6◦ F. She waits 1 hour, takes the temperature of the
body, and finds it to be 93.4◦ F. She also notes that the temperature of the room
is 70◦ F. When was the murder committed?
3.5
The Derivatives of ax and loga x
Theorem 3.11.
Theorem 3.12.
d x
a = (ln a)ax
dx
ah − 1
h→0
h
ln a = lim
Theorem 3.13.
1 1
d
loga x =
dx
ln a x
Exercise 17. Differentiate:
a. y = 2x
b. y = (1.4)x
c. y = 32x
d. y = log8 x
e. y = log x
f. f (x) = log3 (x2 + 1)
g. f (x) = x3 log5 x
3.6
An Economics Application: Elasticity of Demand
Retailers and manufacturers often need to know how a small chance in price will
affect the demand for a product. If a small increase in price produces no change
in demand, a price increase may make sense; if a small increase in price creates
a large drop in demand, the increase is probably ill advised. To measure the
sensitivity of demand to a small percent increase in price, economists calculate
the elasticity of demand.
23
Theorem 3.14. Suppose that q represents a quantity of goods purchased and x
is the price per unit of the goods. Recall that x and q are related by the demand
function q = D(x). Then the elasticity of demand E is given as a function
of price x by
x · D0 (x)
E(x) = −
D(x)
Exercise 18. Klix Video has found that demand for rentals of its DVDs is
given by
q = D(x) = 120 − 20x,
where q is the number of DVDs rented per day at x dollars per rental. Find
each of the following:
a. The quantity demanded when the price is $2 per rental.
b. The elasticity as a function of x.
c. The elasticity at x = 2 and x = 4. Interpret the meaning of these values of
the elasticity.
d. The value of x for which E(x) = 1. Interpret the meaning of this price.
e. The total revenue function, R(x).
f. The price x at which total revenue is a maximum.
Theorem 3.15.
Total revenue is increasing at those x-values for which E(x) < 1.
Total revenue is decreasing at those x-values for which E(x) > 1.
Total revenue is maximized at the x-values for which E(x) = 1.
24
Chapter 4
Integration
4.1
Antidifferentiation
Theorem 4.1. The antiderivative of f (x) is the set of functions F (x) + C
such that
d
[F (x) + C] = f (x).
dx
The constant C is called the constant of integration. If F (x) is antiderivative
of the function f (x), we write
Z
f (x) dx = F (x) + C.
This equation is read as “the antiderivative of f (x), with respect to x, is the
set of functions F (x) + C.” The expression on the left hand side is called an
indefinite integral.
Exercise 1. Determine the indefinite integrals. That is, find the antiderivative
of each integrand:
R
a. 8 dx
R
b. 3x2 dx
R
c. ex dx
R
d. x1 dx
25
Theorem 4.2. Rules of Antdifferentiation
R1. Constant Rule:
Z
k dx = kx + C.
R2. Power Rule:
Z
xn dx =
1
xn+1 + C,
n+1
n 6= −1.
R3. Natural Log Rule
Z
1
dx = ln x + C,
x
R4. Exponential Rule (base e)
Z
1
eax dx = eax + C,
a
x > 0.
a 6= 0.
Exercise 2. Using the rule of antidifferentiation, find the following indefinite
integrals:
R
a. x7 dx
R√
x dx
b.
R 1
c. x3 dx
R
d. e4x dx
Theorem 4.3. Properties of Antidifferentiation
P1. A constant factor can be moved to the front of an indefinite integral:
Z
Z
[c · f (x)] dx = c · f (x) dx.
P2. The antiderivative of a sum or a difference is the sum or the difference of
the antiderivatives:
Z
Z
Z
[f (x) ± g(x)] dx = f (x) dx ± g(x) dx.
Exercise 3. Determine the following indefinite integrals. Assume x > 0.
R
a. (3x5 + 7x2 + 8) dx
R
4
dx
b. 4+3x+2x
x
26
Initial Condition Problem:
Exercise 4. A rock is thrown directly upward with an initial velocity of 50 fps
from an initial height of 10 feet. The velocity of this rock is modeled by the
function v(t) = −32t + 50, where t is in seconds, v is in fps, and t = 0 represents
the moment the rock is released.
a. Determine the distance function h as a function of t. (Why is distance the
same as height?)
b. Determine the height and velocity of the rock after 3 seconds.
4.2
Antiderivatives as Areas
Exercise 5. A vehicle travles at 50 mph for 2 hours. How far has the vehicle
traveled?
Exercise 6. The velocity of a moving object is given by the function v(x) = 3x,
where x is in hours and v is in mph. Use geometry to find the area under the
graph, which is the distance the object has traveled:
a. during the first 3 hours.
b. between the third hour and the fifth hour.
Riemann Summations
We need a method that will allow us to find the area under curved graphs;
so far everything has been a nice graph that we can use simple geometry for.
First, we must introduce some new notation; sigma notation or summation
notation.
Exercise 7. Write the following using sigma notation:
a. 2 + 4 + 6 + 8 + 10
b. 33 + 44 + 55 + 66
c. 5 + 10 + 15 + · · · + 95
d. g(x1 )∆x + g(x2 )∆x + . . . g(x19 )∆x
Exercise 8. Express without using sigma notation:
a.
4
P
3i
i=1
b.
30
P
h(xi )∆x
i=1
27
We use rectangles to approximate the area under a curve given by y = f (x), a
continuous function, over an interval [a, b]. In the figure above, [a, b] is divided
into four subintervals, each having width ∆x = b−a
4 . The heights of the rectangles shown are: f (x1 ), f (x2 ), f (x3 ), and f (x4 ). The area of the region under
the curve is approximately the sum of the areas of the four rectangles:
f (x1 )∆x + f (x2 )∆x + f (x3 )∆x + f (x4 )∆x =
4
X
f (xi )∆x.
i=1
Approximation of area by rectangles becomes more accurate as we use smaller
subintervals and hence more rectangles, as shown in the following figures.
In general, suppose that the interval [a, b] is divided into n equally sized subintervals, each of width ∆x = b−a
n . We construct rectangles with heights
f (x1 ), f (x2 ), . . . , f (xn ). The width of each rectangle is ∆x, so the first rectangle
has an area of f (x1 )∆x, the second rectangle has an area of f (x2 )∆x, and so
on. Thus the area of the region under the curve is approximated by the sum of
the area of the rectangles:
n
X
f (xi )∆x.
i=1
28
Steps for the Process of Riemann Summation
1. Draw the graph of f (x).
2. Subdivide the interval [a, b] into n subintervals of equal width. Calculate
the width of each rectangle by using the formula ∆x = b−a
n .
3. Construct rectangles above the subintervals such that the top left corner
of each rectangle touches the graph.
4. Determine the area of each rectangle.
5. Sum these areas to arrive at an approximation for the total area under
the curve.
Exercise 9. Consider the graph of f (x) = 600x − x2 over the interval [0, 600].
a. Approximate the area by dividing the interval into 6 subintervals.
b. Approximate the area by dividing the interval into 12 subintervals.
Definition 4.1. Let y = f (x) be continuous and nonnegative, f (x) ≥ 0, over
an interval [a, b]. A definite integral is the limit as n → ∞ (equivalently,
∆x → 0) of the Riemann sum of the areas of rectangles under the graph of the
function y = f (x) over the interval [a, b].
Exact area = lim
∆x→0
n
X
Z
f (xi ) · ∆x =
i=1
b
f (x) dx.
a
Make reference to Exercise 6 and 9 using definite integrals this time.
29
4.3
Area and Definite Integrals
Definition 4.2. Let f be any continuous function over the interval [a, b] and
F be any antiderivative of f . Then the definite integral of f from a to b is
Z
b
f (x) dx = F (b) − F (a).
a
Theorem 4.4 (Fundamental Theorem of Calculus). If a continuous function
f has an antiderivative F over [a, b], then
lim
n→∞
n
X
b
Z
f (x) dx = F (b) − F (a).
f (xi )∆x =
a
i=1
Then to find the area under the graph of a nonnegative continuous function f
over the interval [a, b]:
1. Find any antiderivative F (x) of f (x). (Note the lack of C as I show)
2. Evaluate F (x) at x = b and x = a, and compute F (b) − F (a). The result
is the area under the graph of y = f (x) over the interval [a, b].
Exercise 10. Find the area under the curve of
a. y = x2 + 1 over the interval [−1, 2].
b. y =
1
x
c. y =
1
x2
over the interval [1, 4].
over the interval [1, b].
Compare the definite integrals of y = x2 and y = −x2 over the interval [0, 2].
2
x3 8
8
x dx =
= −0=
3 0
3
3
0
Z 2
3 2
x
8
8
−x2 dx = − = − − 0 = −
3
3
3
0
0
Z
2
2
30
Exercise 11. Consider the following integrals and interpret the results:
R2
a. 0 (x2 − 1) dx
R1
b. 0 (x2 − 1) dx
R2
c. 1 (x2 − 1) dx
R1
R2
d. − 0 (x2 − 1) dx + 1 x2 − 1 dx
Exercise 12. The driver of a vehicle traveling at 40 mph (58.67 fps) applies the
brakes, softly at first, then harder, coming to a complete stop after 7 seconds.
The velocity as a function of time is modeled by the function v(t) = −1.197t2 +
58.67, where v is in fps, t is in seconds, and 0 ≤ t ≤ 7. How far did the vehicle
travel while the driver was braking?
4.4
Properties of Definite Integrals
Theorem 4.5. For a < b < c,
Z
c
Z
f (x) dx =
a
b
Z
f (x) dx +
a
c
f (x) dx.
b
For any number b between a and c, the integral from a to c is the integral from
a to b plus the integral from b to c.
Exercise 13. Evaluate each definite integral:
R4
a. −3 |x| dx.
b.
R3
0
|1 − x2 | dx.
31
Suppose you want to find the area of a region bounded above and below by the
graphs of two functions, y = f (x) and y = g(x) as shown below.
Note that the area of this shaded region, A, is given as follows:
A = A2 − A1
Z b
Z
=
f (x) dx −
a
b
g(x) dx
a
b
Z
[f (x) − g(x)] dx
=
a
Theorem 4.6. Let f and g be continuous functions and suppose that f (x) ≥
g(x) over the interval [a, b]. Then the area of the region between the two curves,
from x = a to x = b, is
Z b
[f (x) − g(x)] dx.
a
Exercise 14. Find the area of the region bounded by y = x4 − 3x3 − 4x2 + 10,
y = 40 − x2 , x = 1 and x = 3.
Exercise 15. Find the area of the region bounded by the graphs of f (x) = 2x+1
and g(x) = x2 + 1.
Exercise 16. Find the area of the region bounded by the graphs of f (x) =
x3 + 3x2 − 9x − 12 and g(x) = 4x + 3.
32
Definition 4.3. Let f be continuous function over a closed interval [a, b]. Its
average value, yav , over [a.b] is given by
yav
1
=
b−a
Z
b
f (x) dx.
a
Exercise 17. Find the average value of f (x) = x2 over the interval [0, 2].
4.5
Integration Techniques: Substitution
Consider the following integrals:
Z
Z
x7 dx
(3x − 4)7 dx
But these aren’t the same. Where did the 3 come from? Now we use the method
of substitution.
Exercise 18. Evaluate the following integrals:
R
a. 3x2 (x3 + 1)10
R 2x
b. 1+x
2 dx
c.
R
ln(3x)
x
dx
2
xex dx
R √
e. 5x x2 + 3 dx
d.
R
4.6
Integration Techniques: Integration by Parts
Recall the Product Rule for differentiation:
dv
du
d
(uv) = u
+v .
dx
dx
dx
Integrating both sides with respect to x, we get
Z
Z
dv
du
uv = u dx + v dx
dx
dx
Z
Z
= u dv + v du.
Solving for
R
u dv, we get the following theorem.
33
Theorem 4.7 (Integration by Parts Formula).
Z
Z
u dv = uv − v du
Exercise 19. Evaluate the following integrals:
R
a. xex dx
R
b. ln x dx
R7
c. 0 x2 e−x dx
R
d. x ln x dx
34
Chapter 5
Applications of Integration
setcounterxca0
5.1
An Economics Application: Consumer Surplus and Producer Surplus
The consumer’s demand curve is the graph of p = D(x), which shows the
price per unit that the consumer is willing to pay for x units of a product. It
is usually a decreasing function since the consumer expects to pay less per unit
for large quantities of the product. The producer’s supply curve is the graph
of p = S(x), which shows the price per unit the producer is willing to accept
for selling x units. It is usually an increasing function since a higher price per
unit is an incentive for the producer to make more units available for sale. The
equilibrium point (xE , pE ) is the intersection of these two curves.
Utility is a function often considered in economics; when a consumer receives x units of a product, a certain amount of pleasure, or utility, U is derived
from them.
Definition 5.1. Suppose that p = D(x) describes the demand function for a
commodity. Then the consumer surplus is defined for the point (Q, P ) as
Z
Q
D(x) dx − QP.
0
35
Exercise 20. Find the consumer surplus for the demand function given by
D(x) = (x − 5)2 when x = 3.
Definition 5.2. Suppose that p = S(x) describes the supply function for a
commodity. Then the producer surplus is defined for the point (Q, P ) as
Q
Z
QP −
S(x) dx.
0
Exercise 21. Find the producer surplus for S(x) = x2 + x + 3 when x = 3.
36