Calculus 1 lesson 2 secs 1 3 2 2
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Calculus 1 Lesson 2 Secs 1.3 – 2.2. S. Ives Section 001 MW 1-2:15pm CI-126. Agenda. Section 1.3: Trigonometric Functions Section 1.4: Exponential Functions Sec 1.5: Inverse Functions and Logarithms (Sec 1.6: Graphing with Calculators & Computers) SKIPPING

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Calculus 1 Lesson 2 Secs 1.3 – 2.2

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Calculus 1 lesson 2 secs 1 3 2 2

Calculus 1Lesson 2 Secs 1.3 – 2.2

S. Ives

Section 001

MW 1-2:15pm

CI-126


Agenda

Agenda

  • Section 1.3: Trigonometric Functions

  • Section 1.4: Exponential Functions

  • Sec 1.5: Inverse Functions and Logarithms

  • (Sec 1.6: Graphing with Calculators & Computers) SKIPPING

  • Sec 2.1: Rates of Change and Tangents to Curves

  • Sec 2.2: Limit of a Function and Limit Laws

  • (Sec 2.3: The Precise Definition of a Limit) SKIPPING

  • Homework


Section 1 3 trigonometric functions

Section 1.3: Trigonometric Functions

  • Angles can be thought of as measures of an amount of turn.

  • Angles are measured in degrees or radians.

  • Examples: 30˚= π/6 radians

    135 ˚=(3π)/4 radians

  • In the text angles are measured in radians; keep calculator in radian mode.


6 basic trigonometric functions

6 Basic Trigonometric Functions


Positive or negative

Positive or Negative?

A

All pos

S

sin pos

T

tan pos

C

cos pos


Trig identities transformations

Trig. Identities & Transformations

  • Identities: see text pgs. 26-27

Vertical shift

Vertical stretch or compression

a neg. – reflection about x-axis

Horizontal stretch or compression

b neg. – reflection about y-axis

Horizontal shift


Section 1 4 exponential functions

Section 1.4: Exponential Functions

  • Exponential function with base a:

  • All exponential functions cross the y-axis at 1… why?

  • Recall:


Rules for exponents

Rules for exponents

  • If a>0 and b>0:


Natural exponential function

Natural Exponential Function

  • The graph of has slope 1 when it crosses the y-axis.


Exponential growth decay compound interest

Exponential Growth & Decay; Compound Interest

  • When k>0 it is exponential growth, and when k<0 it is exponential decay. Y0represents the initial amount.

  • Interest compounded continuously: P is the initial investment, r is the interest rate, and t is the time.

  • Example: Use this model to determine the interest on a $100 investment that was invested in 2000 at a rate of 5.5% . How much would be in the account today?

  • y = $173.33


Section 1 5 inverse functions and logarithms

Section 1.5 – Inverse Functions and Logarithms

  • One-to-One Functions: a function that has distinct values at distinct elements in its domain.

  • Definition: A function is one-to-one on a domain D if

  • Horizontal Line Test: a function y = f(x) is one-to-one if and only if its graph intersects each horizontal line at most once.


Inverse functions

Inverse Functions

  • Definition: Suppose f is one-to-one function on a domain D with range R. The inverse function f -1is defined by . The domain of f -1 is R and the range of f -1 is D.

  • 2 step process to finding inverses:

    • 1) Solve y = f(x) for x

    • 2) Interchange x and yIt’s that easy!

  • Example: find the inverse of f(x)=x 3 + 1

  • f -1(x)=3√(x-1)


Logarithmic functions

Logarithmic Functions

  • A logarithmic function is the inverse of an exponential function.

  • Definition: The logarithmic function with base a,

    is the inverse of the base a exponential function (a>0, a≠1)

    Example: natural log. function -


Properties of logarithms

Properties of Logarithms

  • Product Rule:

  • Quotient Rule:

  • Reciprocal Rule:

  • Power Rule:


More properties of logs

More properties of Logs

  • Inverse properties for ax and logax

    • 1. Base a:

    • 2. Base e:

  • Change of Base Formula:


Applications

Applications

  • Why would we ever want to use logs??

  • What if we want to find out how long it will take our savings account to reach a certain amount? How do we get the variable out of the exponent?

  • Example: I have $1000 to invest and want to buy a car for $7000. If I invest it in an account earning 1.73% how long will I have to wait?

  • t=113 years!


More applications

More Applications

  • Since I’ll be dead before I get $7000 let’s try to figure out what rate I need to be able to buy a $7000 in 8 years.

  • 7000 = 1000er8

  • 7 = er8

  • ln 7 = r8

  • r = 0.243 or 24.3%


Chapter 2 limits and continuity

Chapter 2: Limits and Continuity

  • Limits are fundamental to finding velocity and tangents to curves

  • Limits are used to describe the way a function varies:

    • Continuously: small changes in x produce small changes in f(x),

    • Erratic jumps, or

    • Increase or decrease without bound


Section 2 1 rates of change and tangents to curves

Section 2.1: Rates of Change and Tangents to Curves

  • Average Speed is found by dividing the distance covered by the time elapsed.

  • If y is the distance, and t is the time elapsed Galileo’s Law is y = 16t2

  • Average Rate of Change of y = f(x) with respect to x over the interval [x1, x2] is


Illustration of avg rate of change

Illustration of Avg. Rate of Change


Slope of a curve

Slope of a Curve

  • Slope of a curve at a point is the slope of the tangent line

  • To find tangent we look at the limiting behavior of nearby secant lines.

  • Example: find slope of y=x^2 at the point (2,4)

    [see text pg. 58]

  • Instantaneous rate of change is the limit of the average rate of change


Section 2 2 limit of a function and limit laws

Section 2.2: Limit of a Function and Limit Laws

  • Use limits to examine behavior around a point

  • The rest of this section is presented from the text, be sure you understand the theorems:

  • Theorem 1: limit laws

  • Theorem 2: Limits of polynomials

  • Theorem 3: Limits of Rational Functions

  • Theorem 4: The Sandwich Theorem

  • Theorem 5: if f(x)<g(x) then the lim f(x) < lim g(x)


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