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Calculus 1 Lesson 2 Secs 1.3 – 2.2PowerPoint Presentation

Calculus 1 Lesson 2 Secs 1.3 – 2.2

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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

- 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.

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

- Exponential function with base a:
- All exponential functions cross the y-axis at 1… why?
- Recall:

Rules for exponents

- If a>0 and b>0:

Natural Exponential Function

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

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

- 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

- 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

- 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

- Product Rule:
- Quotient Rule:
- Reciprocal Rule:
- Power Rule:

More properties of Logs

- Inverse properties for ax and logax
- 1. Base a:
- 2. Base e:

- Change of Base Formula:

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

- 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

- 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

- 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

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

- 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)