math study guide

1. Functions and Their Properties

• Definition of a Function: Understanding the concept of a function as a relation between inputs and outputs.

• Domain and Range: Identifying the set of possible input values (domain) and output values (range) for functions.

• Function Notation: Utilizing standard notation to represent functions.

Practice Problems:

1. Determine the domain and range of the function f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2​.

2. Evaluate f(3)f(3)f(3) for the function f(x)=2x2−5x+1f(x) = 2x^2 - 5x + 1f(x)=2x2−5x+1.

2. Polynomial Functions

• Definition and Degree: Recognizing polynomial functions and determining their degrees.

• Graphing Polynomials: Plotting polynomial functions and understanding their behavior based on degree and leading coefficients.

• Factoring Polynomials: Breaking down polynomials into products of simpler polynomials.

Practice Problems:

1. Factor the polynomial x3−6x2+11x−6x^3 - 6x^2 + 11x - 6x3−6x2+11x−6.

2. Sketch the graph of f(x)=x3−4xf(x) = x^3 - 4xf(x)=x3−4x.

3. Rational Functions

• Definition: Understanding functions expressed as the ratio of two polynomials.

• Asymptotes: Identifying vertical and horizontal asymptotes in rational functions.

• Graphing Rational Functions: Plotting rational functions, considering asymptotes and intercepts.

Practice Problems:

1. Identify the vertical and horizontal asymptotes of f(x)=2xx2−1f(x) = \frac{2x}{x^2 - 1}f(x)=x2−12x​.

2. Sketch the graph of f(x)=1x−2f(x) = \frac{1}{x - 2}f(x)=x−21​.

4. Exponential Functions

• Definition and Properties: Exploring functions where the variable is in the exponent.

• Graphing Exponential Functions: Understanding the growth and decay behavior of exponential functions.

• Applications: Applying exponential functions to real-world scenarios, such as compound interest and population growth.

Practice Problems:

1. Solve for xxx in the equation 2x=162^x = 162x=16.

2. If a population doubles every 5 years, express the population size as an exponential function of time.

5. Logarithmic Functions

• Definition and Properties: Introducing logarithms as the inverses of exponential functions.

• Laws of Logarithms: Applying logarithmic identities to simplify expressions.

• Solving Logarithmic Equations: Techniques for solving equations involving logarithms.

Practice Problems:

1. Solve log⁡2(x)+log⁡2(x−3)=3\log_2(x) + \log_2(x - 3) = 3log2​(x)+log2​(x−3)=3.

2. Simplify log⁡(a)+log⁡(b)−log⁡(c)\log(a) + \log(b) - \log(c)log(a)+log(b)−log(c).

6. Trigonometric Functions

• Definition and Unit Circle: Understanding sine, cosine, and tangent functions using the unit circle.

• Graphs of Trigonometric Functions: Plotting the basic trigonometric functions and identifying their key characteristics.

• Trigonometric Identities: Utilizing fundamental identities to simplify trigonometric expressions.

Practice Problems:

1. Prove the identity sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1.

2. Solve sin⁡(x)=32\sin(x) = \frac{\sqrt{3}}{2}sin(x)=23​​ for xxx in the interval [0,2π][0, 2\pi][0,2π].

7. Inverse Functions

• Definition: Understanding the concept of inverse functions and their properties.

• Finding Inverses: Techniques to determine the inverse of a given function.

• Graphing Inverse Functions: Plotting inverse functions and understanding their relationship to the original function.

Practice Problems:

1. Find the inverse of f(x)=3x−5f(x) = 3x - 5f(x)=3x−5.

2. Determine if the function f(x)=x2f(x) = x^2f(x)=x2 has an inverse.

8. Systems of Equations

• Linear Systems: Solving systems of linear equations using various methods (substitution, elimination, matrices).

• Nonlinear Systems: Approaches to solve systems involving nonlinear equations.

Practice Problems:

1. Solve the system: {2x+3y=6x−y=4\begin{cases} 2x + 3y = 6 \\ x - y = 4 \end{cases}{2x+3y=6x−y=4​

2. Solve the system: {x2+y2=25y=x+3\begin{cases} x^2 + y^2 = 25 \\ y = x + 3 \end{cases}{x2+y2=25y=x+3​

9. Sequences and Series

• Arithmetic Sequences: Understanding sequences with a constant difference between terms.

• Geometric Sequences: Exploring sequences with a constant ratio between terms.

• Series and Summation: Summing the terms of sequences and understanding convergence.

Practice Problems:

1. Find the 10th term of the arithmetic sequence where the first term is 2 and the common difference is 3.

2. Determine the sum of the first 5 terms of the geometric sequence with a=3a = 3a=3 and r=2r = 2r=2.

10. Conic Sections

• Parabolas, Ellipses, and Hyperbolas: Exploring the equations and properties of conic sections.

• Graphing Conic Sections: Techniques for plotting parabolas, ellipses, and hyperbolas.

Practice Problems:

1. Write the equation of a parabola with vertex at the origin and focus at (0, 2).

2. Determine the center and radii of the ellipse x29+y216=1\frac{x^2}{9} + \frac{y^2}{16} = 19x2​+16y2​=1.

Additional Resources:

• Textbook: The course may utilize resources like Edfinity for assignments and practice problems.

  math.ucsd.edu

• Lecture Notes: Some instructors provide pre-filled lecture notes, which can be invaluable for study and review.

  mathweb.ucsd.edu

• Podcasts: UCSD offers free audio recordings of class lectures, which can be accessed for review.

  podcast.ucsd.edu