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Section 6 WeBWorK

These exercises demonstrate some WeBWorK features.

This problem demonstrates that WeBWorK can process many kinds of answers.

Consider the function \(f\) defined by \(f(x)={\sqrt{x}}\text{.}\)

  1. The exact value of \(f(12)\) is and a decimal approximation for this is .

  2. The domain of this function, in interval notation, is .

  3. The graph of \(y={\sqrt{x}}\) intersects the graph of \(y=6-x\) at .

  4. \(\frac{d}{dx}{\sqrt{x}}={}\) .

  5. The formula for \(f(x)^2\text{,}\) including its restricted domain, is .

  6. \(f\) is a

    • power

    • exponential

    • linear

    • quadratic

    function.

  7. Which is true of the word “radical”?

    • It shares ancestry with "radius", as in the radius of a circle.

    • It shares ancestry with "radish", a vegetable.

    • It shares ancestry with "radler", a mixture of beer and grapefruit soda.

Answer 1.

\(2\sqrt{3}\)

Answer 2.

\(3.4641\)

Answer 3.

\(\left[0,\infty \right)\)

Answer 4.

\(\left(4,2\right)\)

Answer 5.

\(\frac{1}{2\sqrt{x}}\)

Answer 6.

\(x, x\ge 0\)

Answer 7.

\(\text{power}\)

Answer 8.

\(\text{Choice 2}\)

Try multiplying the exponents to see what feedback you get. Also, try something no one should get credit for, like x^2*x^5.

Simplify the expression \({x^{2}x^{5}}\text{.}\)

Hint.

Add the exponents.

Answer.

\(x^{7}\)

Solution.

To simplify the product of two powers of the same base, add the exponents.

\begin{equation*} \begin{aligned} {x^{2}x^{5}}\amp=x^{2+5}\\ \amp={x^{7}} \end{aligned} \end{equation*}

Spelling counts, but not capitalization or spaces.

This Venn Diagram groups animals by certain characteristics.

epspdfpngsvgtex

Name an animal that belongs in the center region. Spelling counts!

Answer.

\({\text{platypus}}\)

WeBWorK has an Open Problem Library with over 40,000 exercises. One of them is this exercise, with file path Library/PCC/BasicAlgebra/NumberBasics/FactorInteger10.pg 1 .

Find the prime factorization of \(35\text{.}\)

\(35={}\)

Answer.

\(5\cdot 7\)

Solution.

After checking to see if small prime numbers divide \(35\text{,}\) we find that \(5\) is one divisor. So \(35=5\cdot7\text{.}\)

Since both \(5\) and \(7\) are prime, the prime factorization of \(35\) is \(5\cdot7\text{.}\)

This problem has multiple parts that must be completed in order. Try answering the second part with various things you might expect a user to enter.

(a) Identify Coefficients.

Consider the equation

\begin{equation*} {2x^{2}-7x-15} = 0 \end{equation*}

Identify the coefficients for the quadratic equation using the standard form from Subsection 1.1.

\(a=\) , \(b=\) , \(c=\)

Answer 1.

\(2\)

Answer 2.

\(-7\)

Answer 3.

\(-15\)

Solution.

Take the coefficient of \(x^2\) for the value of \(a\text{,}\) the coefficient of \(x\) for \(b\text{,}\) and the constant for \(c\text{.}\) In this case, they are \(a = 2\text{,}\) \(b = -7\text{,}\) \(c = -15\text{.}\)

(b) Use the Quadratic Formula.

Use the quadratic formula to find the solution set to

\begin{equation*} {2x^{2}-7x-15}=0 \end{equation*}
Answer.

\(\frac{-3}{2}, 5\)

Solution.

Recall that the quadratic formula is given in Subsection 1.1.

You already identified \(a = 2\text{,}\) \(b = -7\text{,}\) and \(c = -15\text{,}\) and the results from using these in the quadratic formula are \(-\frac{3}{2}\) and \(5\text{.}\)

The answers in this exercise require that units be used.

  1. The average cost of gasoline in the United States in 2010 was $2.78 per gallon. How much gasoline would $20 get you in 2010, on average?

  2. In 2011, the average cost was $3.52 per gallon. What percent increase was that from 2010?

  3. In 2012, the cost had risen 2.8% from the 2011 cost. What was the cost of a gallon of gasoline in 2012?

Answer 1.

\(7.19424\ {\rm gal}\)

Answer 2.

\(26.6\%\)

Answer 3.

\(\$3.62\)

github.com/openwebwork/webwork-open-problem-library/blob/master/OpenProblemLibrary/PCC/BasicAlgebra/NumberBasics/FactorInteger10.pg