Pre-Calculus BC Honors - Midterm Review

This midterm review encapsulates a wide range of topics to prepare students for the upcoming examination. The midterm exam is designed similarly to prior assessments, comprising both calculator-required and no-calculator problems. Students must demonstrate their problem-solving process for most questions, although some will allow for multiple-choice answers without justifications.

General Preparation

Materials to Review:
- Homework assignments
- Class notes
- Past tests and review sheets

Exam Format

Types of Questions:
- No Calculator problems
- Calculator problems
Work Requirement:
- Most problems require showing work
- Some may not require justification or work

No Calculator Problems

Plotting Polar Functions
- Given functions to analyze:
  - a. 𝑟 = 2 cos 𝜃
  - b. 𝑟 = 1 + 2 sin 𝜃
  - c. 𝑟 = 4 − 4 cos 𝜃
- Steps:
  - Plot at least 4 points for each function
  - Sketch the graph of each polar curve
Limits Evaluation
- Evaluate limits algebraically:
  - a. ext{lim}_{x o 3} rac{x^2}{x-1}
  - b. ext{lim}_{x o ext{e}} (x^2 ext{ln} x)
  - c. ext{lim}_{x o 4} (x^2 - 16)/(x-4)
  - d. ext{lim}_{x o 1} (x^3 - 3x^2 + 4x - 1)
  - e. ext{lim}_{x o 0} rac{ an x}{x}
  - f. ext{lim}_{x o -2} (x^2 + 4x + 4)
Piecewise Function Evaluation
- Given function:
  - f(x) = \begin{cases} 5x^2 + 3 & x < -1 \ x & x \geq -1 \end{cases}
- Questions:
  - a. Find f(3) and f(2).
  - b. Determine continuity at x = -1.
Limit Evaluation for Piecewise Functions
- Given function:
  - f(x) = \begin{cases} x - 1 & x \leq 3 \ 2x - 3 & x > 3 \end{cases}
- Find ext{lim}_{x o 3} f(x).
Limit Evaluation at Piecewise Threshold
- Given function:
  - f(x) = \begin{cases} \cos x - \sin \pi & x \leq \pi \ x - \pi - 1 & x > \pi \end{cases}
- Find ext{lim}_{x o 2} f(x).
Continuity Determination
- Given function:
  - f(x) = \begin{cases} 5x^3 + x & x \leq -2 \ -2x^2 & x > -2 \end{cases}
- Determine all x-values where f is continuous.
Function Equation with Asymptotes
- Write an equation for a function with:
  - Removable discontinuities at x = 1 and x = 2
  - Vertical asymptote at x = 3
  - Horizontal asymptote y = 4
Expansion and Evaluation
- Expand and evaluate the following sums using a calculator:
  - Refer to specific examples given in the review.
Trigonometric Identities Verification
- Show that the following identities hold true for all x:
  - a. \sin^3 x + \cos^3 x = 1
  - b. \tan x + \cot x = \sec x \csc x

Calculator Problems

Function Composition
- Given:
  - f(x) = x^2 + 2x - 1 and g(x) = 2x - 3
- Find:
  - a. f(g(x))
  - b. g(f(x))
  - c. g(3x)
  - d. g(f(x) + 2)
Powers of i Simplification
- Simplify:
  - a. i^3
  - b. i^{5}
  - c. i^{-1}
  - d. i^{7}
Simplifying Squares of Negative Numbers
- Simplify:
  - a. \sqrt{-25} \cdot \sqrt{-6}
  - b. \sqrt{-16} \cdot \sqrt{9}
Complex Number Simplifications
- a. (3i - 2) - (-15i - 7)
- b. Simplify expressions involving complex numbers.
Finding Intercepts of Logarithmic Function
- Given the function:
  - y = \log(x - 3) + 2
- Find:
  - a. x-intercepts.
  - b. y-intercepts and sketch.
  - c. Domain of the function.
Determining Domain and Range
- Determine for the following functions:
  - a. a(x) = 3x^2 + 1
  - b. b(x) = -2x^3 - 3x + 1
  - c. c(x) = 2 \ln(x - 2) - 5
  - d. d(x) = 2e^{x} + 3
  - e. e(x) = \sqrt{4 - x}
  - f. f(x) = 3 \sin(2x) + 1
Expansion using Pascal’s Triangle
- a. Expand: (x - 3y)^5
- b. Expand: (2m + 1)^4
Finding Zeros via Synthetic Division
- Given polynomials:
  - a. f(x) = x^3 - 6x^2 + 4x - 24; f(6) = 0
  - b. g(x) = 2x^4 - 7x^3 + 4x^2 + 7x - 6; g(2) = 0
Graphing Polynomials
- Steps to graph the polynomials considering intercepts, behavior at x-intercepts, end behavior using limits:
  - a. For f(x) = x^3 + 4x^2 - 5x
  - b. For g(x) = -x^4 + 4x^2
Permutations vs. Combinations
- Given scenarios decide:
  - a. Selecting 4-person team from 9 athletes - Combination
  - b. Arrangement of letters in “facetious” - Permutation
  - c. Batting lineup of 9 players from 17 - Combination
  - d. Math class sending 2 people to competition - Combination
Hiring Mechanics Calculation
- Given 5 mechanics hired from 8 applicants:
  - Calculate combinations: inom{8}{5}
Song Order Calculation
- Choir practicing 12 songs, ordering 5: Calculate permutations: P(12, 5)
Polynomial Function Factorization with Synthetic Division
- Given function:
  - f(x) = 2x^3 - 4x^2 - 5x - 1
- Perform:
  - a. Synthetic division using factor (x - 3)
  - b. Remainder theorem to evaluate f(3)
  - c. Express rac{f(x)}{(x ext{- factor})} in mixed-number form.
  - d. Find slant asymptote.
Function Evaluations Combining Functions
- Given:
  - f(x) = x^2 + 2 and g(x) = 4x - 1
- Evaluate:
  - a. (f ext{∘} g)(2)
  - b. (g ext{∘} g)(5)
  - c. (g ext{∘} f)(x)
Rational Functions Analysis
- Given rational function:
  - r(x) = \frac{x^2 - 4}{x^2 - 2x}
- Investigate:
  - a. Find discontinuities.
  - b. Vertical asymptote and hole.
  - c. Determine horizontal asymptote.
  - d. Intercepts and graph sketch.
Polynomial Description from Graph Behavior
- Write equations based on:
  - a. x-axis crossing at -2, 1, and 3, with y going to -∞ as x → ∞.
  - b. Bouncing off x-axis at (-1, 0) and (2, 0), with y-intercept at (0, 12).

Multiple Choice Questions

Limit Evaluation
- Evaluate: ext{lim}_{x o 0} rac{x^3}{x^2 + 2}.
- Choices:
  - (A) 0 (B) 1 (C) 4 (D) nonexistent
Limit Evaluation of Quotients
- Evaluate: ext{lim}_{x o 0} rac{x^2}{x^3}{ ext{ln} x}.
  - Choices: (A) 4 (B) 1 (C) 0 (D)
Limit Behavior
- Evaluate: ext{lim}_{x o ext{∞}} rac{1+x}{1+x^2}. Choices: (B) 0 (C) 1 (D)
Asymptotic Behavior Implications
- Given horizontal asymptote y = 2, evaluate conditions.
Limit Existence Analysis
- Analyze graph behavior within domain 0 ≤ x ≤ 4.
Intermediate Value Theorem Application
- Analysis of equations guaranteed solutions on [1, 4].
Zeros Assessment in Continuous Functions
- Determine zeros based on given function values.