Pre-Calculus Quiz Review

Hosted by: Dr. Bruce Piper and TLA Bharath Krishnamurthy

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Overview

• Exponents

• Logarithms

• Unit Circle/Trig Functions

Exponents

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24 = 2 × 2 × 2 × 2 =16Base

Exponent

Simple Exponent Rules to Remember!

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bxby = bx +y

bx

by= bx−y

(bx )y = bxy

Remember these rules only apply toLIKE BASES

Simple Exponent Rules to Remember!

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(bc)x = bxc x

(b

c)x =

bx

c x

Simple Exponent Rules to Remember!

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b1

x = bx

by

x = [(b)y ]1

x

Examples:

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Express as an int value :

64( )2

3

Logarithms

logba = c

bc = a

Logarithmic functions are inverses of exponential

Simple Logarithm Rules to Remember:

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logb (UV ) = logb (U) + logb (V )

logb (U /V ) = logb (U) − logb (V )

logb (U p ) = p logb (U)

logb (1) = 0

logb b =1Pay attention to LIKE bases!!!!

Couple of Side Notes

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blogb x = x

logb bx = x

Couple of Side Notes

‘ln’ is “natural log” which is the SAME thing as logex.

‘e’ is a constant with a value 2.718. It has special meaning because the slope of a tangent line of a function of ‘e’ is the function itself. (Dont worry if you don’t understand this bit)

DON’T GET CONFUSED BY THE TERM ‘ln’. It just means log base ‘e’. You don’t need to memorize the value of ‘e’. You can just express your answer in terms of ‘e’!!

Example Problem:

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5 ⋅4 x = 2x +2

Problems for you to try:

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e ln 34534

1

log8 2

4 x +1 = 164

2x = −4

Unit Circle

Unit Circle

• Circle of Radius 1

• Constructed from pythagoream’s theorem

•Angles correspond to points along the circle: these define our values for cosine and sine

Trigonometric Functions

• From the unit circle we can deduce: • sin(x) = y/r• cos(x) = x/r• tan(x) = y/x• …and of course we all know that x2 + y2 = r2

Example Problems with Right Triangle:

Given that cos (θ) = − 3/ 4 and that sin (θ) > 0, find tan (θ) and express your answer in simplified form (without trigonometric functions).

For some real number x, it is known that sin(x) = ¼ and cos(x) > 0. The value of cos(x) can be written as √α / 4. What is α?

Common Trig Values to Memorize:

Note how sin is ascending and cos is descending

If you have these down, then the tan is simple sin/cos!

Add npi to go to different quadrants on the coordinate axes

AllStudents (sin is positive in 2nd)Take (tan is positive in 3rd) Calculus (cos is positive in 4th)

Inverse Trig Functions

The expression sin(x) = ½ reads “the sine of x is one half”.

The expression sin-1(1/2) = ? is the opposite of the above expression. It can also be interpreted as: “The sine of some angle is ½, what is the angle?”

Example Problem

Evaluate:

tan(sin-1(1/√2))

tan(cot-1(1/3))