Category Archives: MAC1140

Practice Problems

Addition and Multiplication Principles Combined

Solutions to Practice Problems

Addition and Multiplication Principles Combined Solutions

5.1 Exponential Functions, 6.3 Exponential Functions

Exponents: Simplify

July 16, 2018 admin

Example: Rewrite the expression in the form 5^p where p is an algebraic expression.

25/root{3}{5}

Solution:

The original expression
Rewrite the numerator with a base of 5.	$25/root{3}{5}=5^2/root{3}{5}$
Rewrite the radical as a fractional exponent. In general $root{n}{x^m}=x^{m/n}$ . For this example $root{3}{5}= 5^{1/3}$	$5^2/root{3}{5}=5^2/5^{1/3}$
Use the quotient rule for exponents to subtract the exponents. Quotient Rule for Exponents $b^m/b^n=b^{m-n}$	$5^2/5^{1/3}=5^{2-1/3}$
Simplify by getting a common denominator and combining the fractions.	$5^{2-1/3}=5^{6/3-1/3}=5^{5/3}$

3.1 Functions, 3.1 Relations and Functions

Finding Domain: Radical/Root Function (Even Root)

July 12, 2018 admin

Example: Classify the function as a polynomial function, rational function, or root function, and then find the domain. Write the domain interval notation and set builder notation.

f(x) = root{6}{2-x}

Solution:

Classify the Function

Formal Definition	Practical Way of Identifying
Polynomial Function A polynomial function is a function of the form $f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x+a_0$ where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients $a_n , a_{n-1,...,a_1,a_0$ are from the real numbers.	Look for the variables to be in the numerator. (If there is no fraction at all, they are in the numerator.) The variable should not be inside a radical or absolute value. The powers or exponents on the variables should be whole numbers. Whole numbers come from this list {0, 1, 2, 3, 4, …}
Rational Function A rational function is a function of the form where and are polynomial functions and is not equal to zero.	The numerator and denominator are polynomials. Most functions with variables in the denominator are considered Rational Functions but there are exceptions.
Root Function (even index) A root function is a function of the form where n is an even positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an even number. {2, 4, 6, 8, …} The square root is an even index although the index is not written.
Root Function (odd index) A root function is a function of the form where n is an odd positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an odd number. {3, 5, 7, 9, …} The cube root is an odd index.

Since the function f(x) = root{6}{2-x} has a radical and the index is even. This function is a root function.

Find the Domain of a Root Function (Even Index)

Taking the even root of a negative number results in a complex or imaginary number. Since we are interested in real function values, we would like the expression inside the radical to be non-negative ( zero or positive) The root function is defined for any value of the variable where the expression under the radical is non-negative (zero or positive). Find these values by creating an inequality to solve. The inequality is the expression under the radical greater than or equal to zero.

Set the expression under the radical greater than or equal to zero.	0
Solve the inequality. This inequality is a linear inequality and can be solved by isolating the variable on one side.	0
Solve by isolating the variable. Start by subtracting 2 on both sides.
Isolate the variable. Continue by dividing both sides by -1. Be sure to reverse the inequality symbol since you are dividing both sides by a negative.

The function values where x<=2 are defined for f(x) = root{6}{2-x} .

In set builder notation, the domain is .

In interval notation, the domain is ( - infty, 2 ]

1.9 Polynomial and Rational Inequalities, 5.6 Polynomial and Rational Inequalities

Inequalities: Rational Inequalities

July 11, 2018 admin

5.5 Applications of Exponential and Logarithmic Functions, 6.7 Financial Models, Applications of Exponential Functions

Application of Exponential Functions: Doubling Time

July 11, 2018 admin

Example:

How long does it take for an investment to double if it is invested at 18% compounded continuously?

Solution:

Since this question involve continuous compound interest, we will use the associated formula.

$A=Pe^{rt}$

We are given that the interest rate is 18% or 0.18. This tells me that when r=0.18 Since we are looking for the doubling time, A will be 2 times P. I can write that in symbols A=2P.

Substitute these values into the continuous compound formula and solve for the interest rate.

Continuous compound formula	$A=Pe^{rt}$
Substitute the values of r and A into the formula	$A=Pe^{rt}$ $2P=Pe^{0.18*t}$ $2P=Pe^{0.18t}$
Solve for t by dividing both sides by P and simplifying	$2P=Pe^{0.18t}$ ${2P}/P={Pe^{0.18t}}/P$ $2=e^{0.18t}$
Solve for t by taking the log of both sides.	$2=e^{0.18t}$ $ln 2=ln e^{0.18t}$
Solve for t by using the power rule and simplifying	$ln 2=ln e^{0.18t}$
Solve for t by dividing both sides by 0.18 and simplifying
Find the value in the calculator
Write the answer rounded to two decimal places

It will take 3.85 years to double your money when interest is compounded continuously at 18%.

If you need to write this in years and months, you will need to convert the 0.85 to months. Since there are 12 months in a year, multiply 0.85 by 12 to get 10.2. I will round to the nearest months to get 10.

It will take 3 years and 10 months to double your money when interest is compounded continuously at 18%.

Here is a video that is similar except that you are looking for the investment to triple.

5.2 The Real Zeros of a Polynomial Function

Polynomial Function: Finding Zeros and Write in Factored Form

July 11, 2018 admin

Problem: Use the rational zeros theorem to find all real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

f(x)=7x^3-x^2+7x-1

Since f is a polynomial function with integer coefficients use the rational zeros theorem to find the possible zeros.

The factors of the constant term, 1 are p.
p: pm 1

The factors of the leading coefficient, 7 are q.
q: pm 1, pm 7

The possible rational zeros can be found by working out all of the possible combinations of p/q.
p/q : {pm 1}/{pm 1}, {pm 1}/{pm 7}

Simplifying these combinations give p/q : 1, -1, 1/7, -{1/7}

To test if any of these potential zeros are actual zeros, evaluate the function at these values.

x	f(x)	f(x)
-1	7(-1)^3-(-1)^2+7(-1)-1	-16
-1/7	7(-1/7)^3-(-1/7)^2+7(-1/7)-1	-2.041
1/7	7(1/7)^3-(1/7)^2+7(1/7)-1	0
1	7(1)^3-(1)^2+7(1)-1	12

This can be completed quickly using the ask feature in your calculator.

Since f(1/7) is zero, 1/7 is a zero of the function. Since the function has a zero of x=1/7 then the function has a factor of x-1/7

Use long division or synthetic division to to reduce the polynomial.

Write the factor on the outside and the function on the inside of the long division symbol. Make sure both are written in descending order and to use place holders where needed.
Divide the first term of the factor into the first term of the function. Write that value on top of the long division symbol. For this example ${7x^3}/x=7x^2$
Multiply this value by all of the terms in the expression being divided by. Write the terms under the expression you are dividing into and be sure the line up the like terms.
Change the signs of all the terms you just multiplied.
Combine like terms and bring down the next set of terms.
Repeat the process over again. Divide the first term in the factor into the new first term of the function. Write that value on top of the long division symbol. For this example
Multiply this value by all of the terms in the expression being divided by. Write the terms under the expression you are dividing into and be sure the line up the like terms.
Change the signs of all the terms you just multiplied.
Combine like terms. The steps of this process (the division algorithm) are repeated until the degree of the remainder is less than the degree of expression you are dividing by.