Notes

Intended for use with

Calculus with Applications; 10th Edition

ISBN: 0-321-42132-6

1

Derivative Review

Formal Definition of the Derivative: f ‘ (x) =

Example 1: Find f ‘ for f(x) = 3x

2

– 4x + 7

lim

ℎ→0

Constant Rule: if k is a constant, 𝑑 𝑑𝑥

Example 2: 𝑑 𝑑𝑥

Power Rule: 𝑑 𝑑𝑥

[9]

[𝑥 𝑛

] = 𝑛𝑥 𝑛−1

[𝑘] = 0 𝑓(𝑥+ℎ)− 𝑓(𝑥) ℎ

Example 3: 𝑑 𝑑𝑥

Example 4: 𝑑 𝑑𝑥

[𝑥

4

1

√𝑥

]

]

Constant Times a Function: if k is a constant, 𝑑 𝑑𝑥

Sum or Difference Rule: 𝑑 𝑑𝑥

[𝑓(𝑥) ± 𝑔(𝑥)] = 𝑑 𝑑𝑥

[𝑘 ∙ 𝑓(𝑥)] = 𝑘 ∙

[𝑓(𝑥)] ± 𝑑 𝑑𝑥 𝑑 𝑑𝑥

[𝑓(𝑥)] = 𝑘 ∙ 𝑓

[𝑔(𝑥)] = 𝑓

(𝑥) ± 𝑔

(𝑥)

(𝑥)

Example 5: 𝑑 𝑑𝑥

[12𝑥

4

− 6√𝑥 +

5 𝑥

]

2

Product Rule: 𝑑 𝑑𝑥

Example 6: 𝑑 𝑑𝑥

[𝑓(𝑥)𝑔(𝑥)] = 𝑓(𝑥) ∙ 𝑔′(𝑥) + 𝑔(𝑥) ∙ 𝑓′(𝑥)

[(2𝑥 + 3)(3𝑥

2

− 1)] 𝑑

Quotient Rule: 𝑑𝑥

Example 7: 𝑑 𝑑𝑥

[ 𝑓(𝑥)

2𝑥−1

4𝑥+3 𝑔(𝑥)

] =

]

Chain Rule: 𝑑 𝑑𝑥 𝑔(𝑥)∙𝑓′(𝑥)−𝑓(𝑥)∙𝑔′(𝑥)

(𝑔(𝑥))

2

[𝑓(𝑔(𝑥))] = 𝑓 ′(𝑔(𝑥)) ∙ 𝑔

′ 𝑑

(𝑥)

or 𝑑𝑥 or 𝑑 𝑑𝑥

[ ℎ𝑖 𝑙𝑜

] = 𝑙𝑜 𝑑ℎ𝑖−ℎ𝑖 𝑑𝑙𝑜 𝑙𝑜 𝑙𝑜

𝑛

𝑛−1

Example 8: 𝑑 𝑑𝑥

[(𝑥

2

+ 5𝑥)

8

]

3

Example 9: 𝑑 𝑑𝑥

[(2𝑥 + 3)

7

Example 10: 𝑑 𝑑𝑥

[

(3𝑥+2)

7

Exponential Rules: 𝑥−1

(5𝑥 − 6)

8

] 𝑑

1.

2. 𝑑𝑥 𝑑

3. 𝑑𝑥 𝑑 𝑑𝑥 𝑑

4. 𝑑𝑥

𝑥 𝑥

𝑓(𝑥)

𝑓(𝑥) 𝑥

𝑓(𝑥)

𝑥

Example 11: 𝑑 𝑑𝑥

[3𝑒 𝑥

] 𝑓(𝑥)

4

Example 12: 𝑑 𝑑𝑥

[𝑒

5𝑥

]

Example 13: 𝑑 𝑑𝑥

Logarithm Rules:

2𝑥+3

𝑑

1. 𝑑𝑥 𝑑

2. 𝑑𝑥 𝑑

3. 𝑑𝑥 𝑑

4. 𝑑𝑥

𝑎

1 𝑥

[ln|𝑓(𝑥)|] =

[log

|𝑥|] =

𝑎

1 ln 𝑎 𝑓′(𝑥) 𝑓(𝑥)

∙

1 𝑥

|𝑓(𝑥)|] =

1 ln 𝑎

∙ 𝑓′(𝑥) 𝑓(𝑥)

Example 15: 𝑑 𝑑𝑥

[4 ln 𝑥]

Example 16: 𝑑 𝑑𝑥

[ln(𝑥

2

+ 1)]

Example 19: 𝑑 𝑑𝑥

[log

2

(3𝑥

2

− 4𝑥)]

Section 9.1: Functions of Several Variables

Function of Two or More Variables: The expression z = f(x, y) is a function of two variables if a unique value of z is obtained from each ordered pair of real numbers (x, y). The variables x and y are independent variables and z is the dependent variable.

Domain: the set of all ordered pairs of real numbers (x, y) such that f(x, y) exists.

Range: the set of all values of f(x, y)

Example 1: Use f(x, y) = 4x

2

+ 2xy + 3/y to evaluate:

A) f(–1, 3)

5

B) f(2, 0)

C) 𝑓(𝑥+ℎ,𝑦)−𝑓(𝑥,𝑦) ℎ

Plane: The graph of ax + by + cz = d is a plane if a, b, and c are not all 0.

Example 2: Graph the plane: 2x + y + z = 6

Example 3: Graph: x + z = 6

Example 4: Graph z = x

2

+ y

2

6

Name, Equation

Paraboloid

Trace

xy: Point yz: parabola xz: parabola

Graph

Ellipsoid xy: Ellipse yz: ellipse xz: ellipse

Hyperbolic Paraboloid xy: two intersecting lines yz: parabola xz: parabola

Hyperboloid of 2 Sheets xy: none yz: hyperbola xz: hyperbola

7

Section 9.2: Partial Derivatives

Partial Derivatives (Informal Definition):

The partial derivative of f with respect to x,

𝜕𝑧

𝜕𝑥 variable and y as a constant.

, is the derivative of f obtained by treating x as a

The partial derivative of f with respect to y,

𝜕𝑧

𝜕𝑦

, is the derivative of f obtained by treating y as a variable and x as a constant.

Partial Derivatives (Formal Definition): Let z = f(x, y) be a function of 2 independent variables. Let all indicated limits exist. Then a partial derivative of f with respect to x is

8 𝑓

(𝑥, 𝑦) =

𝜕𝑓

= lim 𝑓(𝑥+ℎ,𝑦)𝑐−𝑓(𝑥,𝑦) 𝑥

𝜕𝑥 ℎ →0 ℎ

And the partial derivative of f with respect to y is 𝑓

(𝑥, 𝑦) =

𝜕𝑓

= lim 𝑓(𝑥,𝑦 +ℎ)𝑐−𝑓(𝑥,𝑦) 𝑦

𝜕𝑦 ℎ →0 ℎ

If the indicated limits do not exist, then the partial derivatives do not exist.

Example 1: Let f(x, y) = 4x

2

– 9xy + 6y

3

. Find f x

(x, y) and f y

(x, y).

Example 2: Let f(x, y) = ln |x

2

+ 3y|. Find f x

(x, y) and f y

(x, y).

C)

Example 3: Let f(x, y) = 2x

2

+ 9xy

3

+ 8y + 5

A) Find f x

(–1, 2)

B)

𝜕𝑓

𝜕𝑦

(−4, −3)

All values of x and y such that both f x

(x, y) = 0 and f y

(x, y) = 0

Example 4: Find all second-order partial derivatives for f(x, y) = –4x

3

– 3x

2 y

3

+ 2y

2

9

10

Example 5: Let f(x, y) = 2e x

– 8x

3 y

2

. Find all second-order partial derivatives.

Section 9.3: Maxima and Minima

Relative Maxima and Minima: Let (a, b) be the center of a circular region contained in the xy-plane.

Then, for a function z = f(x, y) defined for every (x, y) in the region,

1. f(a, b) is a relative (or local) maximum if f(a, b) ≥ f(x, y) for all points (x, y) in the circular region

2. f(a, b) is a relative (or local) minimum if f(a, b) ≤ f(x, y) for all points (x, y) in the circular region

Location of Extrema: Let a function z = f(x, y) have a relative maximum or relative minimum at the point (a, b). Let f x

(a, b) and f y

(a, b) both exist. Then f x

(a, b) = 0 and f y

(a, b) = 0.

Example 1: Find all critical points for f(x, y) = 6x

2

+ 6y

2

+ 6xy + 36x – 5.

Test for Relative Extrema: For a function z = f(x, y), let f xx

, f yy

, and f xy

all exist in a circular region contained in the xy-plane with center (a, b). Further, let fx (a, b) = 0 and fy (a, b) = 0.

The discriminant D = 𝑓 𝑥𝑥

(𝑎, 𝑏) ∙ 𝑓 𝑦𝑦

(𝑎, 𝑏) − [𝑓 𝑥𝑦

(𝑎, 𝑏)]

2

1. f(a, b) is a relative maximum if D > 0 and f xx

(a, b) < 0

2. f(a, b) is a relative minimum if D > 0 and f xx

(a, b) > 0

3. f(a, b) is a saddle point (neither a max nor a min) if D < 0

4. if D = 0, the test gives no information.

11

Example 2: In Example 1, f(x, y) = 6x

2

+ 6y

2

+ 6xy + 36x – 5 had a critical number of (-4, 2). Does that point lead to a relative max, a relative min, or neither?

12

Example 3: For f(x, y) = 9xy – x

3

– y

3

– 6, find all points where f has any relative maxima or relative minima.

13

Section 9.4: Lagrange Multipliers

Lagrange Multipliers: All relative extrema of the function z = f(x, y), subject to a constraint g(x, y) =

0, will be found among those points (x, y) for which there exists a value of λ (lambda) such that F x

(x, y, λ) = 0, F y

(x, y, λ) = 0, F

λ

(x, y, λ) = 0 where F(x, y, λ) = f(x, y) – λ ∙ g(x, y) and all partial derivatives exist.

Example 1: Find the minimum value of f(x, y) = 5x

2

+ 6y

2

– xy subject to the constraint x + 2y = 24.

Step 1: Rewrite constraint equal to 0.

Step 2: Form the Lagrange function F(x, y, λ)

Step 3: Find F x

(x, y, λ), F y

(x, y, λ), and F

λ

(x, y, λ)

Step 4: Form F x

(x, y, λ) = 0, F y

(x, y, λ) = 0, F

λ

(x, y, λ) = 0 as a system of equations and solve.

Example 2: Maximize the area, A(x, y) = xy, subject to the constraint xy + 20y + 20x + 474,000 =

500,000

Example 3: Find the dimensions of the closed rectangular box of maximum value that can be produced from 6 square feet of material.

14

Section 9.5: Total Differentials and Approximations

Total Differential for 2 Variables: Let z = f(x, y) be a function of x and y. Let dx and dy be real numbers. Then the total differential of z is dz = f x

(x, y) ∙ dx + f y

(x, y) ∙ dy

Example 1: z = f(x, y) = 9x

3

– 8x

2 y + 4y

3

A) Find dz

B) Evaluate dz when x = 1, y = 3, dx = 0.01, and dy = -0.02.

Approximations: For small values of dx and dy, dz ≈ ∆z where ∆z = f(x + dx, y + dy) – f(x, y).

Example 2: Approximate

√2.98

2

+ 4.01

2

15

16

Approximation by Differentials: For a function f having all indicated partial derivatives, and for small values of dx and dy, f(x + dx, y + dy) ≈ f(x, y) + dz or f(x + dx, y + dy) ≈ f(x, y) + f x

(x, y) ∙ dx + f y

(x, y) ∙ dy

Total Differential for 3 Variables: If w = f(x, y, z), then the total differential dw is dw = f x

(x, y, z) dx + f y

(x, y, z) dy + f z

(x, y, z) dz

Example 3: A short length of blood vessel is in the shape of a right cylinder, where the height is 42 mm ± 0.9 mm (maximum error of 0.9 mm) and the radius is 2.5 mm ± 0.2 mm. Find the maximum possible error in calculating the volume of the blood vessel.

Section 7.1: Antiderivative

Antiderivative: If F ‘ (x) = f(x), then F(x) is an antiderivative of f(x)

Example 1: If F(x) = 10x, then F ‘ (x) = 10, so F(x) = 10x is an antiderivative of f(x) = 10.

Example 2: Find an antiderivative of f(x) = 5x

4

Integration Constant: If F(x) and G(x) are both antiderivatives of f, then there is a constant C such that F(x) – G(x) = C. C is the integration constant and is added to every indefinite integral.

Indefinite Integral: If F ‘ (x) = f(x), then

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶

, read as “the integral of f(x) dx”

Power Rule:

𝑛

𝑥 𝑛+1 𝑛+1

1 𝑛+1

𝑛+1 ,

when n ≠ -1

Example 3:

∫ 𝑥

3 𝑑𝑥

Example 4:

1 𝑥

2

𝑑𝑥

Example 5:

∫ √𝑦𝑑𝑦

Example 6:

∫ 𝑧𝑑𝑧

Constant Rule:

∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶

Example 7:

∫ 17𝑑𝑥

Example 8:

∫ 8𝑑𝑧

Constant Multiple Rule:

∫ 𝑘 ∙ 𝑓(𝑥)𝑑𝑥 = 𝑘 ∙ ∫ 𝑓(𝑥)𝑑𝑥

Example 9:

∫ 2𝑥

5 𝑑𝑥

Example 10:

12 𝑥

3

𝑑𝑥

Sum or Difference Rule:

∫[𝑓(𝑥) ± 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 ± ∫ 𝑔(𝑥)𝑑𝑥

Example 11:

∫(3𝑥

2

− 4𝑥 + 5)𝑑𝑥

Example 12:

𝑥

2

+1

√𝑥

𝑑𝑥

17

Example 13:

∫(𝑥

2

− 1)

2 𝑑𝑥

Exponential Integrals:

𝑥

𝑥

1

𝑘𝑥

𝑥

𝑘𝑥

𝑘

1 ln 𝑎

1 𝑘𝑥

𝑘(ln 𝑎) 𝑥

𝑘𝑥

Example 14:

∫ 9𝑒 𝑥 𝑑𝑥

Example 15:

∫ 𝑒

9𝑥 𝑑𝑥

5

4 𝑥

Example 16:

∫ 3𝑒

Example 17:

∫ 2 𝑥 𝑑𝑥

Example 18:

∫ 3

−5𝑥 𝑑𝑥

Integral of x

-1

:

∫ 𝑥

−1 𝑑𝑥 = ∫ 𝑑𝑥 𝑥

= ln |𝑥| + 𝐶

Example 19:

4 𝑥 𝑑𝑥

Example 20:

∫(−

5 𝑥

+ 𝑒

−2𝑥

)𝑑𝑥

18

Section 7.2: Substitution

Example 1:

∫ 6𝑥(3𝑥

2

+ 4)

7 𝑑𝑥

Rule of thumb 1: If you have a quantity raised to a power, let u = that quantity

Let u = 3x

2

+ 4, then du = 6xdx

3

+ 2)

5 𝑑𝑥

Example 2:

∫ 15𝑥

2

(5𝑥

Example 3:

∫ 𝑥

2

√𝑥

3 𝑥+3

+ 1𝑑𝑥

Example 4: ∫

(𝑥

2

+6𝑥)

2 𝑑𝑥

19

Example 5: ∫

(2𝑥−3)𝑑𝑥

2 𝑥 −3𝑥

Rule of thumb 2: If you have division, let u = the denominator. 𝑥

2

+2𝑥

Example 6: ∫ 𝑥

3

+3𝑥

2 𝑑𝑥

Example 7:

∫ 𝑥

2 𝑒 𝑥

3 𝑑𝑥

Rule of thumb 3: If you have e raised to an exponent, let u = the exponent.

Example 8:

∫ 𝑒

2𝑥+1 𝑑𝑥

Example 9:

∫ 𝑥√1 − 𝑥𝑑𝑥

20

Example 10: ∫ 𝑒

2𝑥 𝑒

2𝑥

+ 5 𝑑𝑥

Example 11:

1

𝑑𝑥

𝑥 ln 𝑥

Section 7.3: Area and the Definite Integral

Definite Integral: If f is defined on the closed interval [a, b], the definite integral of f from a to b is 𝑏

∫ 𝑓(𝑥)𝑑𝑥

= lim 𝑛→∞

∑ 𝑛 𝑖=1 𝑓(𝑥 𝑖

)∆𝑥 𝑎

“the limit as n approaches infinity of the sum from I = 1 to n of f(x i

) times delta x (dx)”

21

Example 1: Approximate

0

4

∫ 2𝑥𝑑𝑥

by using 4 rectangles of equal width whose heights are the midpoints of each subinterval and using geometry.

∆𝑥

(the width of the rectangles) = top endpoint−bottom endpoints number of rectangles

0

4

∫ 2𝑥𝑑𝑥

= ∑

= ∑ 𝑓

4 𝑖=1 𝑖=1 𝑓(𝑥 𝑓(𝑥 𝑖 𝑖

)∆𝑥

)(1)

)(1) + 𝑓(𝑥

= 𝑓(𝑥

1 2

)(1) + 𝑓(𝑥

3

)(1) + 𝑓(𝑥 x

1

is midpoint between 0 and 1, so x

1

= 0.5

4

Similarly, x

2

= 1.5, x

3

= 2.5, and x

4

= 3.5.

= f(0.5) + f(1.5) + f(2.5) + f(3.5)

= 2(0.5) + 2(1.5) + 2(2.5) + 2(3.5)

= 1 + 3 + 5 + 7

= 16

)(1)

=

4−0

4

= 1

Using geometry, first draw the graph, then use the area of a triangle formula.

22

Example 2: Approximate geometry.

0

8

∫ (3𝑥 − 1)𝑑𝑥

by using 4 rectangles of equal width whose heights are A) the right endpoints, B) the left endpoints, C) the midpoints of each subinterval, and D) by using

23

Section 7.4: The Fundamental Theorem of Calculus

Fundamental Theorem of Calculus: Let f be a continuous function on [a, b] and F be any antiderivative of f. Then 𝑎 𝑏

∫ 𝑓(𝑥)𝑑𝑥

= 𝐹(𝑥)| 𝑏 𝑎

= 𝐹(𝑏) − 𝐹(𝑎)

Properties of Definite Integrals:

1.

2.

𝑎 𝑎

∫ 𝑓(𝑥)𝑑𝑥 𝑏

= 0

3.

4.

5.

∫ 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥 𝑎 𝑏 𝑏

∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 𝑐 𝑎 𝑏 𝑎 𝑎 𝑏

, where k is a constant.

= ∫ 𝑓(𝑥)𝑑𝑥

= ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑑𝑥

= − ∫ 𝑓(𝑥)𝑑𝑥 𝑐 𝑏 𝑎 𝑏

± ∫ 𝑔(𝑥)𝑑𝑥

, for any number c between a and b

Example 1:

0

1

∫ 2𝑥𝑑𝑥

=

2 ∙

1

2 𝑥

2

|1

0

(no +C needed)

=

= 𝑥

2

|1

0

[(1)

2

= 1 – 0

= 1

] − [(0)

2

]

On the Calculator:

First, press the MATH button. Scroll down till you find fnInt( and hit ENTER. Now, enter the function you are integrating; here, I enter 2x. Then enter a comma followed by another x, since we are integrating dx. Finally, enter another comma followed by the lower bound, 0, and a third comma followed by the higher bound, 1. Then hit ENTER to get your answer.

Example 2:

2

5

∫ (6𝑥

2

− 3𝑥 + 5)𝑑𝑥

Example 3:

0

5

∫ 𝑥√25 − 𝑥

2 𝑑𝑥

24

Example 4: Find the area of the region between the x-axis and the graph of 𝑓(𝑥) = 𝑥

2 x = 1 to x = 3.

− 3𝑥

from

Section 7.5: Area Between Two Curves

Area between 2 curves: If f and g are continuous functions and f(x) ≥ g(x) on the interval [a, b], then the area between f and g from x = a to x = b is given by: 𝑏

𝐴 = ∫ [𝑓(𝑥) − 𝑔(𝑥)]𝑑𝑥 𝑎

Example 1: Find the area bounded by

(𝑥) = −𝑥

2

+ 1

,

Example 2: Find the area between 𝑦 = 𝑥

1/2 and 𝑦 = 𝑥

Example 3: Find the area between y = 2x and y = x

3

3 𝑔(𝑥) = 2𝑥 + 4

, x = –1, and x = 2.

Section 7.6: Numerical Integration

Trapezoidal Rule: Let f be a continuous function on [a, b] and let [a, b] be divided into n equal subintervals by the points a = x

0

, x

1

, x

2

, …, x n

= b. Then 𝑎 𝑏

≈ ( 𝑏−𝑎

) [ 𝑛

1

2 𝑓(𝑥

0

) + 𝑓(𝑥

1

) + ⋯ + 𝑓(𝑥 𝑛−1

) +

1

2 𝑓(𝑥 𝑛

)]

Example 1: Use the trapezoidal rule with n = 4 to approximate

0

2

2

+ 1𝑑𝑥

Simpson’s Rule: Let f be a continuous function on [a, b] abd ket [a, b] be divided into an even number n of equal subintervals by the points a = x

0

, x

1

, x

2

, …, x n

= b. Then 𝑏

𝑎

0

1

2

3

𝑛−2

𝑛−1

𝑛

)]

Example 2: Use Simpson’s Rule with n = 4 to approximate

2

∫ √𝑥

2

25

Section 8.1: Integration by Parts

Integration by Parts: If u and v are differentiable functions, then

∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢

Example 1:

∫ 𝑥𝑒

5𝑥 𝑑𝑥

Example 2:

∫ 𝑥𝑒

−2𝑥 𝑑𝑥

Example 3:

∫ ln 𝑥 𝑑𝑥

, x > 0

Example 4:

∫ ln 2𝑥 𝑑𝑥

26

Example 5:

∫(2𝑥

2

Example 6:

∫(3𝑥

2 ln 𝑥

+ 5)𝑒

−3𝑥 𝑑𝑥

+ 4)𝑒

2𝑥 𝑑𝑥

Example 7:

1 𝑒 𝑥

2 𝑑𝑥

Example 8:

1

4− 𝑥

2

𝑑𝑥

27

Example 9: ∫

1 𝑥√4+ 𝑥

2 𝑑𝑥

Section 8.2: Volume and Average Value

Volume of a Solid of Revolution: If f(x) is nonnegative and R is the region between f(x) and the xaxis from x = a to x = b, the volume of the solid formed by rotating R about the x-axis is:

V = lim

∆𝑥→0

∑ 𝑛 𝑖=1 𝜋[𝑓(𝑥 𝑖

)]

2

∆𝑥 𝑎 𝑏

2 𝑑𝑥

Example 1: Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by y = x + 1, y = 0, x = 1, and x = 4.

Example 2: Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by y = x2 + 1, y = 0, x = –1, and x = 1.

28

29

Example 3: Find the volume of the solid of revolution formed by rotating about the x-axis the area bounded by f(x) = 4 – x

2

and the x-axis.

Average Value of a Function: Over the interval [a, b] is

1 𝑏−𝑎 𝑎 𝑏

∫ 𝑓(𝑥)𝑑𝑥

Example 4: Find the average value of the function f(x) = 25 – 5e

-0.01x

over [0, 6]

Section 8.3: Continuous Money Flow

Exponential Growth Equation: 𝑦(𝑡) = 𝑦

0 𝑒 𝑘𝑡 amount, and y(t) is the amount after time t.

, where k is the rate, t is time, y

0

(“y-not”) is the initial

Example 1: The income from a soda machine (in dollars per year) is growing exponentially. When the machine was first installed, it was producing income at a rate of \$500 per year. By the end of the first year, it was producing income at a rate of \$510.10 per year. Find the total income produced by the machine during its first 3 years of operation.

30

Total Money Flow: If f(x) is the rate of money flow, then the total money flow over the time interval from t = 0 to t = T is:

0

𝑇

∫ 𝑓(𝑡)𝑑𝑡

Present Value of Money Flow: If f(x) is the rate of continuous money flow at an interest rate r for T years, then the present value is P

=

𝑇

−𝑟𝑡

𝑑𝑡

Example 2: A company expects its rate of annual income during the next three years to be given by f(t) = 75,000t, 0 ≤ t ≤ 3. What is the present value of this income over the 3-year period, assuming an annual interest rate of 8% compounded continuously?

Accumulated Amount of Money Flow at Time T: If f(x) is the rate of money flow at an interest rate

B) r at time t, the accumulated amount of money flow at time T is A

6% interest compounded continuously, find:

A) the total money flow over the 5-year period

=

𝑟𝑇

0

𝑇

−𝑟𝑡

𝑑𝑡

Example 3: If money is flowing continuously at a constant rate of \$2,000 per year over 5 years at the accumulated amount of money flow at time T = 5

B)

C) the total interest earned

D) the present value of the amount with interest.

Example 4: A continuous money flow starts at a rate of \$1,000 per year and increases exponentially at 2% per year.

A) Find the accumulated amount of money flow at the end of 5 years at 10% interest compounded continuously.

Find the present value at 5% interest compounded continuously.

Section 8.4: Improper Integrals

Improper Integrals: If f is continuous on the indicated interval and if the indicated limit exists, then

∞ 𝑏

−∞

−∞

𝑏→ ∞

𝑎→ −∞ 𝑐

−∞ 𝑏

𝑏

𝑐

Example 1:

1

∞ 𝑑𝑥 𝑥

31

32

Example 2: ∫

8

1

Example 3

:

−∞ 𝑥

1

3

𝑑𝑥

−3𝑥

0

Example 4:

−2𝑥

𝑑𝑥

Section 10.1: Solutions of Elementary and Separable Differential Equations

Example 1: Find all the solutions of the differential equation: 𝑑𝑦 𝑑𝑥

= 3𝑥

2

− 2𝑥

General Solution of 𝑑𝑦 𝑑𝑥

= 𝑔(𝑥):

The general solution is y =

∫ 𝑔(𝑥)𝑑𝑥

Example 2: 𝑑𝑃

= 20𝑒

0.05𝑡 where P is the bird population and t is time in years. Find P in terms of t 𝑑𝑡 if there were 20 birds initially.

Particular Solution: A problem where a given value can be used to find C.

Example 3: Find the particular solution of 𝑑𝑦 𝑑𝑥

− 2𝑥 =

2𝑥

√𝑥

2

+3 given that y(1) = 8.

Example 4: Find the particular solution of 𝑑𝑦 𝑑𝑥

Example 5: Find the general solution of 𝑦 𝑑𝑦 𝑑𝑥

− 12𝑥

= 𝑥

2

3

= 6𝑥

2 given that y(2) = 60.

33

Example 6: Find the general solution of 𝑑𝑦 𝑑𝑥

=

𝑥

2

+1 𝑥𝑦

2

Section 10.2: Linear First-Order Differential Equations

Linear First-Order Differential Equation: of the form 𝑑𝑦 𝑑𝑥

Example 1: Solve: 𝑥 𝑑𝑦 𝑑𝑥

+ 6𝑦 + 2𝑥

4

= 0

+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)

Integrating Factor: The function

𝐼(𝑥) = 𝑒

∫ 𝑃(𝑥)𝑑𝑥

is called an integrating factor for the differentiable equation 𝑑𝑦 𝑑𝑥

+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)

Solving a Linear First-Order Differentiable Equation:

1. Put the equation in the linear form 𝑑𝑦 𝑑𝑥

+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)

2. Find the integrating factor

𝐼(𝑥) = 𝑒

∫ 𝑃(𝑥)𝑑𝑥

3. Multiply each term of the equation from Step 1 by I(x)

4. Replace the sum of terms on the left with

5. Integrate both sides

6. Solve for y

𝐷 𝑥

[𝐼(𝑥)𝑦]

34

35

Example 2: Give the general equation for: 𝑑𝑦 𝑑𝑥

+ 2𝑥𝑦 = 𝑥

Example 3: Give the general equation for: 𝑥 𝑑𝑦

− 𝑦 − 𝑥

2 𝑒 𝑥

= 0 𝑑𝑥

Example 4: Solve the initial value problem

2 𝑑𝑦

− 6𝑦 − 𝑒 𝑥

= 0

with y(0) = 5. 𝑑𝑥

Section 10.4: Applications of Differential Equations

Continuous Deposits: 𝑑𝐴

= 𝑟𝐴 + 𝐷

, where A is the amount of money invested, r is the interest rate, 𝑑𝑡 t is time in years, and D is the amount deposited per year.

36

Example 1: If \$5000 in initially invested and \$1000 is deposited continuously into an account earning 8% interest, how much will be made after 18 years?

Lotka-Volterra Predator-Prey Equation: 𝑑𝑦

𝑝𝑦−𝑞𝑥𝑦

=

𝑦(𝑝−𝑞𝑥)

, where x is the predator 𝑑𝑥 −𝑟𝑥+𝑠𝑥𝑦 𝑥(𝑠𝑦−𝑟) population, y is the prey population, py is rate the prey population grows if there are no predators, qxy is the rate the predators consume prey, rx is the rate the predators would decreased if there was no prey, and the rate of growth of the predators is sxy.

Example 2: Using the Lotka-Volterra equation, let p = 3, q = 1, r = 4, and s = 2. Suppose that at the time where there are 100 predators (x = 100), there are 1000 prey (y = 1000). Find an equation relating x and y.

37

Section 11.1: Continuous Probability Models

Probability Function of a Random Variable: If the function f is a probability function with domains

{x

1

, x

2

, …, x n

} and f(x i

) is the probability that event x i

occurs, then for 1 ≤ i ≤ n, 0 ≤ f(x i

) ≤ 1 and f(x

1

)

+ f(x

2

) + … + f(x n

) = 1

Continuous Probability Distribution: A continuous random variable can take on any value in some interval of real numbers. The distribution of this random variable is a continuous probability distribution.

Probability Density Function: The function f is a probability distribution function of a random variable X in the interval [a, b] if:

1. f(x) ≥ 0 for all x in the interval [a, b]

2. 𝑏

∫ 𝑓(𝑥)𝑑𝑥

= 1

Example 1: Show that the function defined by f(x) = (3/26)x

2

is a probability distribution function for [1, 3]. The find the probability that X will be between 1 and 2.

Example 2: Is there a constant k such that f(x) = kx

2

is a probability density function over [0, 4]?

Probability Density Functions on (-∞, ∞): If f is a probability density function for a continuous variable X on (-∞, ∞),

𝑃(𝑋 ≤ 𝑏) = 𝑃(𝑋 < 𝑏) = ∫ 𝑏 𝑓(𝑥)𝑑𝑥

𝑃(−∞ < 𝑋 < ∞) = ∫

−∞ 𝑓(𝑥)𝑑𝑥 = 1

Example 3: Suppose a random variable X is the distance in kilometers from a given point to the nearest bird’s nest, with the probability density function of the distribution given by

2𝑥𝑒

−𝑥

2

for x ≥ 0. Show that f(x) is a probability density function and find the probability that there is a bird’s nest within 0.5 km of the given point.

Section 12.1: Geometric Sequences

General Term of a Geometric Sequence:

𝑛

= 𝑎𝑟

𝑛−1

Example 1: Find the first 4 terms of the sequence a n

= –4n + 2

38

Example 6: Find the sum:

∑

4 𝑖=0

39

B)

Example 2: Find the indicated term for each geometric sequence:

A) Find a

7

for 6, 24, 96, 384, …

Find a

6

for 8, –16, 32, –64, 128, …

Sum of the First n Terms of a Geometric Sequence: If a geometric sequence has the first term a and a common ratio r, then the sum of the first n terms, S n

, is:

𝑛 𝑛

𝑎(𝑟

= ∑

𝑛

−1)

, where r ≠ 1 𝑟−1 𝑛−1 𝑖=0

𝑎𝑟

𝑖

Example 4: Find the sum of the first 6 terms of the geometric sequence 3, 12, 48, …

Example 5: Find the sum:

8 𝑖=0

1

2

𝑖

1

3

)

𝑖

40

Section 12.2: Annuities: An Application of Sequences

Amount of Annuity: The amount S of an annuity of payments of R dollars each, made at the end of each period for a consecutive interest periods at a rate of interest I per year, is

𝑆 = 𝑅 [

(1+𝑖) 𝑛 𝑖

−1

] 𝑜𝑟 𝑆 = 𝑅 ∙ 𝑠 𝑛̅|𝑖

“s-angle-n at I”

Example 1: If someone deposits \$22,000 at the end of each year for 7 years in an account paying

8% compounded annually, how much money will they have after the 7 years?

Example 2: Suppose \$1,000 is deposited at the end of each 6-month period for 5 years in an account paying 6% per year compounded semiannually. Find the amount of the annuity.

Example 3: How much money must be deposited yearly into a sink fund so that \$17,000 is available after 5 years, at a rate of 8% compounded annually?

Present Value of an Annuity: The present value P of an annuity of payments of R dollars each, made at the end of each period for n consecutive interest periods at a rate of interest I per period is:

1− (1+𝑖)

−𝑛

𝑃 = 𝑅 [ 𝑖

] 𝑜𝑟 𝑃 = 𝑅𝑎 𝑛̅|𝑖

Example 4: What lump sum deposited today at 6% interest compounded annually will yield the same total amount as payments of \$1500 at the end of each year for 12 years, also at 6% compounded annually?

Section 12.3: Taylor Polynomials at 0

Factorial: a! “a factorial” = a ∙ (a – 1) ∙ (a – 2) ∙ … ∙ (3) ∙ (2) ∙ (1)

Taylor Polynomial for f(x) = e x

: The Taylor polynomial of degree n for f(x) = e x

at x = 0 is

𝑃 𝑛

(𝑥) = ∑ 𝑛 𝑖=0 𝑓

(𝑖) 𝑖!

(0) 𝑥 𝑖

= ∑ 𝑛 𝑖=0

1 𝑖!

𝑥 𝑖

Example 1: Use a Taylor polynomial of degree 5 to approximate e

–0.2

Taylor Polynomial of Degree n: Let f be a function that can be differentiated n times at 0. The

Taylor polynomial of degree n for f at 0 is

𝑃 𝑛

(𝑥) = 𝑓(0) + 𝑓

(1)

(0)

1!

𝑛 𝑖=0 𝑓

(𝑛)

(0) 𝑖!

𝑥 𝑖 𝑥 + 𝑓

(2)

(0)

2!

𝑥

2

+ 𝑓

(3)

(0)

3!

𝑥

3

+ ⋯ + 𝑓

(𝑛)

(0) 𝑛!

𝑥 𝑛

=

41

42

Example 2: Let 𝑓(𝑥) = √𝑥 + 1

. Find the Taylor polynomial of degree 4 at x = 0. Then approximate

√0.9

Example 3: Let

𝑓(𝑥) =

1

1−𝑥

. Find the Taylor polynomial of degree n at x = 0. The approximate

1/0.98.

Section 12.4: Infinite Series

Infinite Series: an expression of the form

1

2

3

𝑛

+ ⋯ = ∑

Example 1: Find the first 5 partial sums for the sequence 1, 1/2, 1/4, 1/8, 1/16, …

∞ 𝑖=0

𝑎

𝑖

43

Sum of the Infinite Series: Let

𝑆 𝑛 𝑎

1

+ 𝑎

2

+ 𝑎

3

+ ⋯ + 𝑎 𝑛 sum of the infinite series

= 𝑎

1

+ 𝑎

2

+ ⋯

. Suppose lim 𝑛→∞ 𝑎

1

+ 𝑎

2

+ 𝑎

3

+ 𝑎

𝑆 𝑛

3

+ ⋯ + 𝑎

+ ⋯

, and the infinite series converges. If no such limit exists, then the infinite sum has no sum and diverges. 𝑛

be the nth partial sum for the series

= 𝐿

for some real number L. Then L is called the

+ ⋯ + 𝑎 𝑛

+ 𝑎𝑟

3

+ ⋯

converges, if r is Sum of a Geometric Series: the infinite geometric series 𝑎 + 𝑎𝑟 + 𝑎𝑟 in (–1, 1), to the sum 𝑎

1−𝑟

. The series diverges if r is not in (–1, 1).

Example 2: If the series converges, find the sum.

A)

3

8

3

64

3

512

2

B)

3

9

27

81

+ ⋯

4 16 64 256

C) 1 + 1.1 + (1.1)

2

+ (1.1)

Section 12.5: Taylor Series

3

+ (1.1)

4

+ … (1.1) n

+ …

Taylor Series: If all derivatives of a function f exist at 0, then the Taylor Series for f at 0 is: 𝑓(0) + 𝑓

(1)

(0𝑥) + 𝑓

(2)

(0) 𝑥

2

+ 𝑓

(3)

(0)

2!

3!

Example 1: Find the Taylor Series for f(x) = e x

at 0. 𝑥

3

+ ⋯

44

C)

B)

Example 2: Find the Taylor Series for each function:

A) 𝑓(𝑥) = 5𝑒 𝑥 𝑓(𝑥) = 𝑥

3 ln(1 + 𝑥)

−𝑥

2

2

45

D)

E)

1

1+4𝑥

1

2− 𝑥

2

46