Section 3.4

Mar 30, 2019

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Section 3.4. Properties of Logarithms. Properties of Logs. EXAMPLES:. Use exponential notation. Raise both sides to the power of c. Use properties of exponents. Take the log base b of both sides. Simplify left side. “Back substitute” for x. Change of Base Formula.

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Section 3.4

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Section 3.4 Properties of Logarithms
Properties of Logs EXAMPLES:
Use exponential notation Raise both sides to the power of c Use properties of exponents Take the log base b of both sides Simplify left side “Back substitute” for x
Change of Base Formula This means you CAN evaluate logs with ANY base!
Write in exponential notation Take log base a of both sides Use Power Rule Dived both sides by “Back substitute” for x

Section 3.4

Section 3.4. Recursive Definitions. Recursion. Recursion is the process of defining an object in terms of itself Technique can be used to define sequences, functions and sets To recursively define a sequence: give first term give rule for defining subsequent terms from previous terms.

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Section 3.4

Section 3.4. The Traveling Salesperson Problem. By Aaron Desrochers and Ben Epstein. Tucker Applied Combinatorics. To 1 2 3 4 From 1 3 9 7 2 3 6 5 3 5 6 6 4 9 7 4. 4 1. 3 2.

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Section 3.4

Section 3.4. Objectives: Find function values Use the vertical line test Define increasing, decreasing and constant functions Interpret Domain and Range of a function Graphically and Algebraically. * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9. x * z *

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Section 3.4

Rational Function. A rational function is a function of the formWhere P and Q are polynomials and P(x) andQ(x) have no factor in common and Q(x) is notequal to zero.. Example. Consider . Find the domain and graph f.Solution: When the denominator x 4 = 0, we have x = ?4, so the only input that results in a denominator of 0 is ?4. Thus the domain is {x|x ? ?4} or (??, ?4) ? (?4, ?)..

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Section 3.4

Section 3.4. Types of Proofs. Theorems. Thm 3.3: If two lines are perpendicular, then they intersect to form four right angles. Thm 3.4: All right angles are congruent. Thm 3.5: If two lines intersect to form a pair of adjacent congruent angles, then the lines are perpendicular.

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Section 3.4

Section 3.4. Arc length & area of a sector. Arc Length. Arc Length. Arc Length. Arc Length. To use this formula, must be in radians. Arc Length. #1: Find the radius of an 15 foot arc of a circle that subtends and angle of 15˚. To use this formula, must be in radians. Arc Length.

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Section 3.4

Section 3.4. Applications. Example 1. A coin bank holds nickels, dimes, and quarters. There are 45 coins in the bank and the value of the coins is $4.75. If there are five more nickels than quarters, find the number of each type of coin in the bank . N = # of nickels D = # of dimes

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Section 3.4

Section 3.4. 1 st Day. Read p. 190. Test for Concavity. Let f be a function whose second derivative exists on an open interval I . 1. If f ꞌꞌ( x ) > 0 for all x in I , then the graph of f is concave upward in I .

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Section 3.4

Section 3.4. Cross Product. THE CROSS PRODUCT. If u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ) are vectors in 3-space, then the cross product or u × v is defined by u × v = ( u 2 v 3 − u 3 v 2 , u 3 v 1 − u 1 v 3 , u 1 v 2 − u 2 v 1 ) or in determinant notation,.

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Section 3.4

Section 3.4. Speed of a wave. The Speed of a Wave. A wave is a disturbance that moves along a medium from one end to the other . The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel.

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Section 3.4

Section 3.4. Additional Topics in Probability and Counting. Section 3.4 Objectives. Determine the number of ways a group of objects can be arranged in order Determine the number of ways to choose several objects from a group without regard to order

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Section 3.4

Chapter 3. Section 3.4. Exercise #5. +. =. =. 1. =. 3. Add. +. =. Chapter 3. Section 3.4. Exercise #11. +. +. =. =. =. Add. +. +. +. +. Chapter 3. Section 3.4. Exercise #17. –. =. =. Subtract. –. =. –. =. Chapter 3. Section 3.4. Exercise #19. –. =. =.

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Section 3.4

Section 3.4. Rational Functions. Rational Function. A rational function is a function of the form Where P and Q are polynomials and P(x) and Q(x) have no factor in common and Q(x) is not equal to zero. Consider . Find the domain and graph f . Solution:

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Section 3.4

Section 3.4. Linear Programming. y – x + y x + y 3 x – 11. 2 3. 11 3. <. 1 4. 11 4. >. >. Step 1 : Graph the constraints. 2 3. 11 3. y = – x + y = x +. To find A , solve the system .

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Section 3.4

Section 3.4. Slope and Rates of Change. Page 190. Slope. The rise , or change in y, is y 2  y 1 , and the run , or change in x, is x 2 – x 1. Example. Page 191. Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. Solution. ( –4 , 1 ).

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Section 3.4

Section 3.4. Library of Functions; Piecewise-Defined Functions. THE SQUARE ROOT FUNCTION. The x -intercept is 0. The y -intercept is also 0. The function is neither even nor odd. It is increasing on the interval (0, ∞). It has a local minimum of 0 at x = 0. Properties of.

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Section 3.4

Section 3.4. Day 2. When is it appropriate to use a least squares regression line (LSRL) or correlation?. When is it appropriate to use a least squares regression line (LSRL) or correlation? Only when you have a linear pattern. Linear Pattern or Not?. Linear Pattern or Not?. No…

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Section 3.4

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Section 3.4

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Section 3.4

Section 3.4. Diagnostics: Looking for Features that the Summaries Miss. Influential Data Points. Some data points have more influence than others in determining where the LSRL goes or on the size and sign of the correlation. Which points do we need to be concerned about?.

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