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Calculus 1 Lesson 2 Secs 1.3 – 2.2

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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  1. Calculus 1Lesson 2 Secs 1.3 – 2.2 S. Ives Section 001 MW 1-2:15pm CI-126

  2. 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

  3. 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.

  4. 6 Basic Trigonometric Functions

  5. Positive or Negative? A All pos S sin pos T tan pos C cos pos

  6. 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

  7. Section 1.4: Exponential Functions • Exponential function with base a: • All exponential functions cross the y-axis at 1… why? • Recall:

  8. Rules for exponents • If a>0 and b>0:

  9. Natural Exponential Function • The graph of has slope 1 when it crosses the y-axis.

  10. 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

  11. 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.

  12. 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 y It’s that easy! • Example: find the inverse of f(x)=x 3 + 1 • f -1(x)=3√(x-1)

  13. 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 -

  14. Properties of Logarithms • Product Rule: • Quotient Rule: • Reciprocal Rule: • Power Rule:

  15. More properties of Logs • Inverse properties for ax and logax • 1. Base a: • 2. Base e: • Change of Base Formula:

  16. 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!

  17. 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%

  18. 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

  19. 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

  20. Illustration of Avg. Rate of Change

  21. 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

  22. 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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