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YOUSSEF FRANCIS

MATH 250

Calculus 1 Lecture notes

Southwestern College Chula Vista

Not returnable

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• 1 Functions Contents
  • 1.1 Review of Functions
  • 1.2 Representing Functions
  • 1.3 Trigonometric Functions
• 2 Limits
  • 2.1 The Idea of Limits
  • 2.2 Definitions of Limits
  • 2.3 Techniques for Computing Limits
  • 2.4 Infinite Limits
  • 2.5 Limits at Infinity
  • 2.6 Continuity
  • 2.7 Precise Definitions of Limits
• Extra Practice Exercises–Set
• 3 Derivatives
  • 3.1 Introducing the Derivative and 3.2 The Derivative as a Funtion
  • 3.3 Rules of Differentiation
  • 3.4 The Product and Quotient Rules
  • 3.5 Derivatives of Trigonometric Functions
  • 3.6 Derivatives as Rates of Change
  • 3.7 The Chain Rule
  • 3.8 Implicit Differentiation
  • 3.9 Related Rates
• Extra Practice Exercises–Set
• 4 Applications of the Derivative
  • 4.1 Maxima and Minima
  • 4.2 Mean Value Theorem
  • 4.3 What Derivatives Tell Us
  • 4.4 Graphing Functions
  • 4.5 Optimization Problems
  • 4.6 Linear Approximation and Differentials
  • 4.9 Antiderivatives
• Extra Practice Exercises–Set
• 5 Integration
  • 5.1 Approximating Areas under Curves
  • 5.2 Definite Integrals
  • 5.3 Fundamental Theorem of Calculus
  • 5.4 Working with Integrals
  • 5.5 Substitution Rule
• Extra Practice Exercises–Set
• 7 Logarithmic and Exponential Functions
  • 7.1 Inverse Functions
  • 7.2 The Natural Logarithmic and Exponential Functions
  • 7.3 Logarithmic and Exponential Functions with Other Bases
• 6 Applications of Integration
  • 6.2 Regions Between Curves
  • 6.3 Volume by Slicing
  • 6.4 Volume by Shells
• Extra Practice Exercises–Set

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Definition: Composite Functions

Given two functions f and g, the composite function f ◦ g is defined by

f ◦ g

(x) = f

g(x)

Note: f

g(x)

6 = g

f (x)

unless f (x) and g(x) are inverse functions in which case f

g(x)

= g

f (x)

= x.

Ex 2. Given f (x) = 3x^2 − x and g(x) = 2x + 1. Find the following. Simplify your answer. i) f (5x + 1).

ii) (f ◦ g)(x).

iii) (g ◦ f )(x).

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Definition: Difference Quotient:

Given a function f (x), the slope formula f (x + h) − f (x) h

is known as the difference quotient.

Ex 3. Let f (x) =

x + 1 and g(x) = x^2 − 3 x − 2. Find the following. i) The difference quotient of f (x).

ii) The difference quotient of g(x).

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iii) h(x) =

x^3 − x

iv) f (x) = 3x^3 − 2 x

v) g(x) = x x^2 + 4

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1.2 Representing Functions

Ex 1. Find the x and y-intercepts of y = 2x −

x^2 + 1.

Ex 2. Find the x and y-intercepts then graph f (x) =

−x^2 + 4 2 x − 3

Ex 3. Graph: f (x) =

2 x + 5 if x ≤ − 1 −x + 3 if x > − 1

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Ex 5. Consider f (x) =

x^2 + x − 2 (x + 1)(x + 2)

i) Find the domain of f (x).

ii) Find the roots of f (x).

Ex 6. Find all the points of intersection of x^2 − y = 3 and x − y = 1.

Ex 7. Find all the points of intersection of x^2 + y^2 = 25 and y = x − 1.

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1.3 Trigonometric Functions

Radian Measure: Calculus typically requires that angles be measured in radians (rad).

Working with a circle of radius r, the radian measure of an angle θ is the length of the arc associated with θ denoted S, divided by the radius of the circle r.

S

r θ

Trigonometric Functions:

• sin θ =

O

H

• cos θ =

A

H

• tan θ =

O

A

• csc θ =

H

O

• sec θ =

H

A

• cot θ =

A

O

Adjacent side(A)

Opposite side(O)

Hypotenuse(H)

θ

Trigonometric Identities:

sin(−θ) = − sin(θ). cos(−θ) = cos(θ).

Pythagorean Identities:

sin^2 θ + cos^2 θ = 1 1 + cot^2 θ = csc^2 θ tan^2 θ + 1 = sec^2 θ

Double- and Half-Angle Formulas

sin 2θ = 2 sin θ cos θ cos 2θ = cos^2 θ − sin^2 θ

cos^2 θ =

1 + cos 2θ 2

sin^2 θ =

1 − cos 2θ 2

Period of Trigonometric Functions: The functions sin θ, cos θ, sec θ, and csc θ have a period of 2π for all θ in the domain.

sin(θ + 2π) = sin θ sec(θ + 2π) = sec θ cos(θ + 2π) = cos θ csc(θ + 2π) = csc θ

The functions tan θ and cot θ have a period of π for all θ in the domain.

tan(θ + π) = tan θ cot(θ + π) = cot θ,

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ii) Given sec θ =

and

3 π 2

< θ < 2 π Find cos θ, sin θ, tan θ, csc θ, and cot θ.

Ex 3. Solve for θ. i) 2 θ cos θ + θ = 0.

ii) sin^2 θ − 1 = 0.

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iii) 2 sin^2 θ + 3 cos θ − 3 = 0.

iv) cot θ cos^2 θ =

cot θ.

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Tangent line s(t) = − 16 t^2 + 96t

t

s(t)

Ex 2. Let s(t) =

2 − t

. Complete the following table and then make a conjecture about the instantaneous veolicty

at t = 0s.

Time Interval Average Velocity

[0, 1]

[0, 0 .1]

[0, 0 .01]

[0, 0 .001]

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2.2 Definitions of Limits

Definition: Limit of a Function (Preliminary): Suppose the function f is defined for all x near a except possibly at a. If f (x) is arbitrarily close to L (as close to L as we like) for all x sufficiently close (but not equal) to a, we write lim x→a f (x) = L and say the limit of f (x) as x approaches a equals L.

Ex 1. Use the graph of f in the figure to find the following values or state that they do not exist:

1 2 3 4 5

1

2

3

4

y = f (x)

x

y (^) • f (1) =

• lim x→ 1 f (x) =
• f (3) =
• lim x→ 3 f (x) =

Definition: One-Sided Limits

1. Right-sided limit: Suppose f is defined for all x near a with x > a. If f (x) is arbitrarily close to L for all x sufficiently close to a with x > a, we write lim x→a+^ f (x) = L and say the limit of f (x) as x approaches a from the right equals L.
2. Left-sided limit: Suppose f is defined for all x near a with x < a. If f (x) is arbitrarily close to L for all x sufficiently close to a with x < a, we write lim x→a−

f (x) = L and say the limit of f (x) as x approaches a from the left equals L.

Ex 2. Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.

y = f (x)

x

y (^) • f (1) =

• lim x→ 1 −

f (x) =

• lim x→ 1 +^

f (x) =

• lim x→ 1

f (x) =

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Ex 5. Create a table of values of f (x) =

x − 1 x − 1 corresponding to values of x near 1. Then make a conjecture about

the value of lim x→ 1 f (x).

x 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.

f (x) =

x − 1 x − 1

Ex 6. Create a table (and a graph) of values of f (x) =

x^3 − 8 4 x − 2

corresponding to values of x near

. Then make a

conjecture about the value of lim x→ 1 / 2

f (x).

x

f (x) =

x^3 − 8 4 x − 2

Ex 7. Create a table (and a graph) of values of f (x) =

x^2 − 9 2 x − 6 corresponding to values of x near 3. Then make a

conjecture about the value of lim x→ 3 f (x).

x

f (x) = x^2 − 9 2 x − 6

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Ex 8. There are times when the table method fails; for instance fill the following table of values and make a conjecture

about the lim x→ 0 cos

x

x y = cos

x

x y = cos

x

To see this better let x =

nπ

then cos x = cos

nπ

= cos(nπ) =

1 if n is even

− 1 if n is odd

x y = cos

x

π

1 2 π

1 3 π

1 4 π

1 5 π

.. .

x y = cos

x

π

2 π

3 π

4 π

5 π

.. .