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# Calculus 1 Lesson 2 Secs 1.3 – 2.2 - PowerPoint PPT Presentation

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 1Lesson 2 Secs 1.3 – 2.2

S. Ives

Section 001

MW 1-2:15pm

CI-126

• 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

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

• Angles are measured in degrees or radians.

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

A

All pos

S

sin pos

T

tan pos

C

cos pos

• 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

• Exponential function with base a:

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

• Recall:

• If a>0 and b>0:

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

• 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

• 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.

• 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)

• 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 -

• Product Rule:

• Quotient Rule:

• Reciprocal Rule:

• Power Rule:

• Inverse properties for ax and logax

• 1. Base a:

• 2. Base e:

• Change of Base Formula:

• 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!

• 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

• 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 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

• 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)