Question 1 of 20
| Functions - Applied
· Level 3
Find a formula for the described function and state its domain.
A rectangle has a perimeter of 20 m. Express the area of the rectangle as a function of the length of one of its sides.
Question 2 of 20
| Functions - Domain
· Level 3
Find the domain of the function.
\(h(x) = \dfrac{1}{\sqrt[4]{x^2 - 5 x}}\)
Question 3 of 20
| Functions - Definition
· Level 2
Determine whether the equation defines \(y\) as a function of \(x\).
\((y + 3)^3 + 1 = 2 x\)
Question 4 of 20
| Functions - Applied
· Level 3
Find a formula for the described function and state its domain.
An open rectangular box with volume 2 m\({}^3\) has a square base. Express the surface area of the box as a function of the length of a side of the base.
Question 5 of 20
| Functions - Piecewise
· Level 2
Evaluate \(f(-3)\), \(f(0)\), and \(f(2)\) for the piecewise defined function. Then sketch the graph of the function.
\( f(x) = \begin{cases} -1 & \text{if } x \leq 1 \\ 7 - 2 x & \text{if } x > 1 \end{cases} \)
Question 6 of 20
| Functions - Definition
· Level 1
Determine whether the table defines \(y\) as a function of \(x\).
| \(x\) (Height in inches) | 72 | 60 | 60 | 63 | 70 |
|---|---|---|---|---|---|
| \(y\) (Shoe size) | 12 | 8 | 7 | 9 | 10 |
Question 7 of 20
| Functions - Evaluation
· Level 2
If \(g(x) = \dfrac{x}{\sqrt{x + 1}}\), find \(g(0)\), \(g(3)\), \(5 g(a)\), \(\dfrac{1}{2} g(4 a)\), \(g(a^2)\), \([g(a)]^2\), \(g(a + h)\), and \(g(x - a)\).
Question 8 of 20
| Functions - Absolute Value
· Level 2
Sketch the graph of the function.
\(g(t) = |1 - 3 t|\)
Question 9 of 20
| Functions - Absolute Value
· Level 2
Sketch the graph of the function.
\(f(x) = |x + 2|\)
Question 10 of 20
| Functions - Difference Quotient
· Level 3
Evaluate the difference quotient for the given function. Simplify your answer.
\(f(x) = 4 + 3 x - x^2\), \(\quad \dfrac{f(3 + h) - f(3)}{h}\)
Question 11 of 20
| Limit Laws - Absolute Value
· Level 3
Find the limit, if it exists. If the limit does not exist, explain why.
\(\operatorname*{lim}\limits_{x \rightarrow 0^-} \left(\dfrac{1}{x} - \dfrac{1}{|x|}\right)\)
Question 12 of 20
| Limit Laws - Algebraic
· Level 2
Evaluate the limit, if it exists.
\(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{x^2 - x - 6}{3 x^2 + 5 x - 2}\)
Question 13 of 20
| Limit Laws - Algebraic
· Level 2
Evaluate the limit, if it exists.
\(\operatorname*{lim}\limits_{x \rightarrow 4} \dfrac{x^2 + 3 x}{x^2 - x - 12}\)
Question 14 of 20
| Limit Laws - Direct Evaluation
· Level 2
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).
\(\operatorname*{lim}\limits_{t \rightarrow -1} \left(\dfrac{2 t^5 - t^3}{5 t^2 + 4}\right)^3\)
Question 15 of 20
| Limit Laws - Algebraic
· Level 2
Evaluate the limit, if it exists.
\(\operatorname*{lim}\limits_{x \rightarrow -5} \dfrac{2 x^2 + 9 x - 5}{x^2 - 25}\)
Question 16 of 20
| Limit Laws - Applied/Proof
· Level 3
Show by means of an example that \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) g(x)]\) may exist even though neither \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) nor \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)\) exists.
Question 17 of 20
| Limit Laws - Applied/Proof
· Level 5
The figure shows a fixed circle \(C_1\) with equation \((x - 1)^2 + y^2 = 1\) and a shrinking circle \(C_2\) with radius \(r\) and center the origin. \(P\) is the point \((0, r)\), \(Q\) is the upper point of intersection of the two circles, and \(R\) is the point of intersection of the line \(P Q\) and the \(x\)-axis. What happens to \(R\) as \(C_2\) shrinks, that is, as \(r \rightarrow 0^+\)?
Question 18 of 20
| Limit Laws - Piecewise/Special
· Level 3
(a) If \(\lfloor x \rfloor\) denotes the greatest integer function, evaluate:
(i) \(\operatorname*{lim}\limits_{x \rightarrow -2^+} \lfloor x \rfloor\)
(ii) \(\operatorname*{lim}\limits_{x \rightarrow -2} \lfloor x \rfloor\)
(iii) \(\operatorname*{lim}\limits_{x \rightarrow -2.4} \lfloor x \rfloor\)
(b) If \(n\) is an integer, evaluate:
(i) \(\operatorname*{lim}\limits_{x \rightarrow n^-} \lfloor x \rfloor\)
(ii) \(\operatorname*{lim}\limits_{x \rightarrow n^+} \lfloor x \rfloor\)
(c) For what values of \(a\) does \(\operatorname*{lim}\limits_{x \rightarrow a} \lfloor x \rfloor\) exist?
Question 19 of 20
| Limit Laws - Algebraic
· Level 2
(a) What is wrong with the following equation?
\(\dfrac{x^2 + x - 6}{x - 2} = x + 3\)
(b) In view of part (a), explain why the equation
\(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 + x - 6}{x - 2} = \operatorname*{lim}\limits_{x \rightarrow 2} (x + 3)\)
is correct.
Question 20 of 20
| Limit Laws - Absolute Value
· Level 3
Find the limit, if it exists. If the limit does not exist, explain why.
\(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{2 - |x|}{2 + x}\)
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