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Pre-Calculus - Term 2 Assignment – Ms. Han shan@daltonschool.kr HS Room 214 INTERES

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Pre-Calculus
- Term 2 Assignment
–
Ms. Han shan@daltonschool.kr HS Room 214
INTEREST PACKET Exponential and Logarithmic Functions; Trigonometric Functions
Students will study a class of functions called exponential functions. Exponential functions are used in modeling many real-world phenomena, such as the growth of a population or the growth of an investment that earns compound interest. Once an exponential model is obtained, we can use the model to predict population size or calculate the amount of an investment for any future date. Students will also study the trigonometric functions using the unit circle. The trigonometric functions can be defined in two different but equivalent ways: as functions of real numbers or as functions of angles. Students will study both approaches to solve different applications that require different approaches.
Topic Overview
Exponential Functions
The Natural Exponential Function
Logarithmic Functions
Laws of Logarithms
Exponential and Logarithmic Equations
Modeling with Exponential and Logarithmic Functions
The Unit Circle
Trigonometric Functions of Real Numbers
Trigonometric Graphs
More Trigonometric Graphs
Inverse Trigonometric Functions and Their Graphs
Angle Measure
Trigonometry of Right Triangles
Trigonometric Functions of Angles
Inverse Trigonometric Functions and Triangles
The Law of Sines
The Law of Cosines
Essential Questions
What is the exponential function?
What is the inverse of the exponential function from both a numeric and geometric perspective?
Define angle of elevation.
Is it possible for two different angles to have the same reference angle?
Is it possible to find the area of a triangle if the side lengths are known, but the angles are not?
Reference Materials
J. Stewart, L. Redlin and S. Watson,
Precalculus: Mathematics for Calculus, Sixth Edition
. (2012: Cengage Learning, Belmont, CA)
Skills Mastery Check
Interpreting Functions
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (10.IF.7.e)
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (10.IF.9)
Building Functions
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (10.BF.5)
Linear, Quadratic, and Exponential Models
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. (10.LQEM.1.a)
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Interpret expressions for functions in terms of the situation they model. (10.LQEM.4)
Trigonometric Functions
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (10.TF.1)
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. (10.TF.2)
Use special triangles to determine geometrically the values of sine, cosine, tangent for p/3, p/4 and p/6, and use the unit circle to express the values of sine, cosines, and tangent for x, p + x, and 2p
–
x in terms of their values for x, where x is any real number. (10.TF.3)
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (10.TF.5)
Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (10.TF.6)
Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. (10.TF.7)
Prove the Pythagorean identity sin2(x) + cos2(x) = 1 and use it to find sin(x), cos(x), or tan(x) given sin(x), cos(x), or tan(x) and the quadrant of the angle. (10.TF.8)
Use the Law of Sines to solve triangles and understand which cases are ambiguous and which cases are unambiguous. (10.TF.9)
Use the Law of Cosines and use Heron
’
s Formula to find the area of a triangle. (10.TF.10)
LESSON OVERVIEW
Day Topics Learning Objectives
Students will be able to …
Homework
1 4.1 Exponential Function
To define an exponential function including what it means to raise
a
to an irrational number. (10.LQEM.1.a) 2 4.2 The Natural Exponential Function
To study the special base
e
which is convenient for applications involving calculus. (10.LQEM.4) 3 4.3 Logarithmic Functions
To study the inverse of exponential functions. (10.IF.7.e) 4 4.4 Laws of Logarithms
To study properties of logarithms. These properties give logarithmic functions a wide range of applications. (10.BF.5) 5 4.5 Exponential and Logarithmic Equations
To solve equations that involve exponential or logarithmic functions. The techniques that students develop here will be used to solve applied problems. (10.IF.9) 6 4.6 Modeling with Exponential and Logarithmic Functions
To model with exponential and logarithmic functions. (10.IF.9) 7 Chapter 4 Review 8
Chapter 4 Quiz
9 5.1 The Unit Circle
To explore some properties of the circle or radius 1 centered at the srcin. (10.TF.1) 10 5.2 Trigonometric Functions of Real Numbers
To use properties of the unit circle to define the trigonometric functions. (10.TF.2) 11 5.3 Trigonometric Graphs
To graph the sine and cosine functions and certain transformations of these functions. (10.TF.5) 12 5.4 More Trigonometric Graphs
To graph the tangent, cotangent, secant, and cosecant functions and transformations of these functions. (10.TF.6) 13 5.5 Inverse Trigonometric Functions and Their Graphs
To graph the inverse trigonometric functions. (10.TF.7) 14 Chapter 5 Review
15
Chapter 5 Quiz
16 6.1 Angle Measure
To define conterminal angles, arc lengths, and linear and angular speed. (10.TF.1) 17 6.2 Trigonometry of Right Triangles
To define six trigonometric functions as ratios of sides of right triangles and solve applications that involve solving right triangles. (10.TF.3) 18 6.3 Trigonometric Functions of Angles
To find the reference angle for a given angle and use the reference angle to evaluate trigonometric functions. (10.TF.8)
19 6.4 Inverse Trigonometric Functions and Triangles
To define and evaluate the inverse sine, cosine, and tangent functions. (10.TF.7) 20 6.5 The Law of Sines
To use the Law of Sines to solve triangles and understand which cases are ambiguous and which cases are unambiguous. (10.TF.9) 21 6.6 The Law of Cosines
To use the Law of Cosines to solve triangles and
to use Heron’s Formula to find the area of a
triangle. (10.TF.10) 22 Final Review
*The schedule above is tentative. Please check the classroom board and Google classroom for the updates.
Expansion Pack:
Please speak to me if you are interested in additional exercises.

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