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Questions and Answers
Which of the following refers to a set where each input corresponds to exactly one output?
Which of the following refers to a set where each input corresponds to exactly one output?
- function (correct)
- relation
- equation
- graph
Find the domain of the function $f(x) = \sqrt{(4-x)} + (\frac{1}{\sqrt{(x^2-1)}})$.
Find the domain of the function $f(x) = \sqrt{(4-x)} + (\frac{1}{\sqrt{(x^2-1)}})$.
- $(-\\infty, -1) \\cup (1, 4)$ (correct)
- $(-4, -1) \\cup (1, \\infty)$
- $(-\\infty, -4) \\cup (-1, 1]$
- $(-\\infty, -1] \\cup (-4, 1)$
Find the domain of the function $f(x) = \ln(x^2-5x+6)$.
Find the domain of the function $f(x) = \ln(x^2-5x+6)$.
- $(-\\infty, -2) \\cup (3, +\\infty)$
- $(-\\infty, -3) \\cup (-2, +\\infty)$
- $(-\\infty, 2) \\cup (3, +\\infty)$
- $(-\\infty, -3) \\cup (2, +\\infty)$ (correct)
What is the limit of the function $f(x) = \frac{x^2-5x+6}{x-3}$ as x approaches 3?
What is the limit of the function $f(x) = \frac{x^2-5x+6}{x-3}$ as x approaches 3?
What is the limit of the function $f(x) = \frac{x^2-5x+6}{x-3}$ as x approaches 5?
What is the limit of the function $f(x) = \frac{x^2-5x+6}{x-3}$ as x approaches 5?
Find the limit of $ (2 - x) \tan( \pi x) $ as x approaches 1.
Find the limit of $ (2 - x) \tan( \pi x) $ as x approaches 1.
Find the limit of $\sin(2(x-\frac{\pi}{4}))/ (x-\frac{\pi}{4}) $ as x approaches $\frac{\pi}{4}$.
Find the limit of $\sin(2(x-\frac{\pi}{4}))/ (x-\frac{\pi}{4}) $ as x approaches $\frac{\pi}{4}$.
Evaluate $\lim_{x \to 0} \frac{1}{x} \int_2^{2+x} (t+\sqrt{t^2+5}) dt$.
Evaluate $\lim_{x \to 0} \frac{1}{x} \int_2^{2+x} (t+\sqrt{t^2+5}) dt$.
What is the limit of (2-x)tan(Ï€x) as x approaches 1?
What is the limit of (2-x)tan(Ï€x) as x approaches 1?
Find the limit of sin(2(x-π/4))/(x-π/4) as x approaches π/4.
Find the limit of sin(2(x-π/4))/(x-π/4) as x approaches π/4.
Evaluate the limit: lim_{x→0} ∫(t+√(t^2+5)) dt/x^2
Evaluate the limit: lim_{x→0} ∫(t+√(t^2+5)) dt/x^2
Given a linear piecewise function f(x) = {x-4 for x<5, x^2 for x≥5}, solve for lim_{x→5^-} f(x).
Given a linear piecewise function f(x) = {x-4 for x<5, x^2 for x≥5}, solve for lim_{x→5^-} f(x).
What type of discontinuity is present at x=2 for the function f(x)= (x^2-4)/(x-2)?
What type of discontinuity is present at x=2 for the function f(x)= (x^2-4)/(x-2)?
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Study Notes
Functions and Relations
- A function is a mathematical relation that relates an input from a set called the domain to exactly one output in a set called the range.
- A function can be represented as a relation, equation, or graph.
Domain of Functions
- The domain of a function is the set of input values for which the function is defined.
- The domain of a function can be found by analyzing the function's formula.
Limits of Functions
- The limit of a function as x approaches a certain value is the value that the function approaches as x gets arbitrarily close to that value.
- The limit of a function can be determined using various techniques, such as factoring, canceling, or using trigonometric identities.
Piecewise Functions
- A piecewise function is a function that is defined differently for different intervals of the input variable.
- The limit of a piecewise function can be evaluated by considering the limit of each piece of the function separately.
Continuity of Functions
- A function is continuous at a number a if the function is defined at a and the limit of the function as x approaches a exists and is equal to the value of the function at a.
- The conditions for continuity of a function at a number a are:
- f(a) is defined
- limx→af(x)lim_{x \to a} f(x)limx→a​f(x) exists
- limx→af(x)=f(a)lim_{x \to a} f(x) = f(a)limx→a​f(x)=f(a)
Calculus Problems
Problem 6: Limit of (2-x)tan(2) as x approaches 1
- The problem involves finding the limit of an algebraic function multiplied by a trigonometric function
- The function is (2-x)tan(2) and the limit is as x approaches 1
- Possible answers: A. e^2, B. e^2/π, C. 0, D. ∞
Problem 7: Limit of sin(2(x-π/4))/(x-π/4) as x approaches π/4
- The problem involves finding the limit of a trigonometric function
- The function is sin(2(x-π/4))/(x-π/4) and the limit is as x approaches π/4
- Possible answers: A. 0, B. 1, C. 2, D. ½
Problem 8: Evaluating a Limit
- The problem involves evaluating a limit of an integral
- The integral is ∫t+√(t^2+5)t 1/(x^2) dx and the limit is as x approaches 0
- Possible answers: A. 2, B. 3, C. 4, D. 5
Problem 9: Solving for A in a Piecewise Function
- The problem involves solving for a parameter in a piecewise function
- The function is f(x)={x-4, x and the goal is to solve for A
- No possible answers are provided
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