Chapter 1 & 2 Test: Answer Key and Review
Welcome to Professor Baker's Math Class! This post provides the answer key to the Chapter 1 and 2 test, covering essential concepts in Calculus 1. Use this resource to solidify your understanding and prepare for future assessments. Let's get started!
Key Concepts Covered
- Difference Quotient: Understanding and evaluating the difference quotient is crucial for grasping the concept of the derivative. Remember the formula: $\frac{f(a+h)-f(a)}{h}$. This represents the average rate of change of a function over an interval.
- Function Classification: Identifying functions as power, polynomial, or algebraic functions is fundamental. Polynomial functions are sums of terms of the form $ax^n$, where n is a non-negative integer.
- Function Composition: The composition of functions, denoted as $(f \circ g)(x) = f(g(x))$, involves plugging one function into another. Pay close attention to the domain of the composite function.
- Transformations of Functions: Shifting graphs horizontally and vertically are essential skills. For example, the graph of $y = e^{x-4}$ is the graph of $y = e^x$ shifted 4 units to the right.
- Inverse Functions: Finding the inverse function involves swapping $x$ and $y$ and solving for $y$. The inverse of $f(x)$ is denoted as $f^{-1}(x)$.
- Limits: Understanding the concept of a limit is crucial to calculus. We explore limits graphically and numerically.
- Evaluating Limits: Techniques include direct substitution, factoring, rationalizing, and using limit laws. Remember that $\lim_{x \to a} f(x)$ exists if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.
- Continuity: A function $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$. Understand different types of discontinuities (removable, jump, infinite).
- Asymptotes: Identifying horizontal and vertical asymptotes is crucial for sketching functions. Vertical asymptotes occur where the denominator of a rational function is zero, and horizontal asymptotes describe the function's behavior as $x$ approaches infinity.
Sample Problems and Solutions
Let's look at some examples from the test:
- Question 1: Difference Quotient
Given $f(x) = x^2 - x + 4$, find the difference quotient. The solution involves substituting $(a+h)$ and $a$ into the function, simplifying, and dividing by $h$, leading to $2a + h - 1$. - Question 3: Function Composition
If $f(x) = \frac{x-1}{x}$ and $g(x) = \frac{x}{x+3}$, find $(f \circ g)(x)$. The solution involves substituting $g(x)$ into $f(x)$ and simplifying to get $\frac{-3}{x}$. Don't forget to consider the domain! - Question 11: Limits by Factoring
Evaluate $\lim_{x \to -2} \frac{x^2 - x - 6}{x + 2}$. By factoring the numerator, we get $\frac{(x+2)(x-3)}{x+2}$, which simplifies to $x-3$. Therefore, the limit is $-2 - 3 = -5$. - Question 14: Limits by Rationalization
Find $\lim_{x \to 9} \frac{3 - \sqrt{x}}{x - 9}$. Rationalize the numerator by multiplying by $\frac{3 + \sqrt{x}}{3 + \sqrt{x}}$. This simplifies to $\frac{-1}{3 + \sqrt{x}}$, and the limit as $x$ approaches 9 is $\frac{-1}{6}$.
Remember to practice these types of problems to build your confidence. Good luck with your studies, and keep up the great work!