Search
- Filter Results
- Location
- Classification
- Include attachments
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/01%3A_Analyzing_Functions/1.04%3A_Function_Families/1.4.03%3A_Graphing_Cube_Root_FunctionsFor \(\ y=\sqrt[3]{x}\), the output is the same as the input of \(\ y=x^{3}\). Starting with \(\ y=\sqrt[3]{x}\), you would obtain \(\ y=\sqrt[3]{x+4}-11\) by shifting the function to the left four un...For \(\ y=\sqrt[3]{x}\), the output is the same as the input of \(\ y=x^{3}\). Starting with \(\ y=\sqrt[3]{x}\), you would obtain \(\ y=\sqrt[3]{x+4}-11\) by shifting the function to the left four units and down 11 units. The general equation for a cubed root function is \(\ f(x)=a \sqrt[3]{x-h}+k\), where h is the horizontal shift and k is the vertical shift.
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/07%3A_Sequences_Series_and_Mathematical_Induction/7.04%3A_Sums_of_Geometric_Series/7.4.02%3A_Sums_of_Infinite_Geometric_SeriesWe know that the pieces have to add up to some finite time period (no matter what it feels like, Sayber CAN get the room clean), but how is it possible for the sum of an infinite number of terms to be...We know that the pieces have to add up to some finite time period (no matter what it feels like, Sayber CAN get the room clean), but how is it possible for the sum of an infinite number of terms to be a finite number? In terms of the actual sums, what is happening is this: as n increases, the n th term gets smaller and smaller, and so the n th term contributes less and less to the value of S n . We say that the series converges, and we can write this with a limit:
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/07%3A_Sequences_Series_and_Mathematical_Induction/7.07%3A_Geometric_Sequences/7.7.02%3A_Finding_the_nth_Term_Given_Two_Terms_for_a_Geometric_SequenceWe will be using the general rule for the \(\ n^{t h}\) term in a geometric sequence and the given term(s) to determine the first term and write a general rule to find any other term. Equation 2: \(\ ...We will be using the general rule for the \(\ n^{t h}\) term in a geometric sequence and the given term(s) to determine the first term and write a general rule to find any other term. Equation 2: \(\ a_{10}=\frac{1}{4}\), so \(\ \frac{1}{4}=a_{1} r^{9}\), solving for \(\ a_{1}\) we get \(\ a_{1}=\frac{\frac{1}{4}}{r^{9}}\). Find the common ratio and the \(\ n^{t h}\) term rule for the geometric sequence given that \(\ a_{1}=-\frac{16}{625}\) and \(\ a_{6}=-\frac{5}{2}\).
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/01%3A_Analyzing_Functions/1.01%3A_Relations_and_Functions
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/03%3A_Exponential_and_Logarithmic_Functions/3.03%3A_Properties_of_Logarithms/3.3.02%3A_Power_Property_of_LogarithmsBecause the expression within the natural log is in parenthesis, start with moving the 4 th power to the front of the log. Now, let's condense log 9 − 4 log 5 − 4 log x + 2 log 7 + 2 log y. log 9 − 4 ...Because the expression within the natural log is in parenthesis, start with moving the 4 th power to the front of the log. Now, let's condense log 9 − 4 log 5 − 4 log x + 2 log 7 + 2 log y. log 9 − 4 log 5 − 4 log x + 2 log 7 + 2 log y log 9 − log 5 4 − log x 4 + log 7 2 + log y 2 =\log _{16} x^{2}+\log _{16} y-\left(\log _{16} 32+\log _{16} z^{5}\right) \\ Equivalent logs are: \(\ \log 25+8 \log c, \log 25+\log c^{8} \text { and } \log 25 c^{8}\).
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/03%3A_Exponential_and_Logarithmic_Functions/3.05%3A_Exponential_and_Logarithmic_Models/3.5.04%3A_The_Number_eThe interest on a sum of money that compounds continuously can be calculated with the formula I=Pe rt −P, where P is the amount invested (the principal), r is the interest rate, and t is the amount of...The interest on a sum of money that compounds continuously can be calculated with the formula I=Pe rt −P, where P is the amount invested (the principal), r is the interest rate, and t is the amount of time the money is invested. The value of Steve’s car decreases in value according to the exponential decay function: V=Pe −0.12t , where V is the current value of the vehicle, t is the number of years Steve has owned the car and P is the purchase price of the car, $25,000.
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/05%3A_Vector_Analysis/5.04%3A_Vector_Equations_and_Applications
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/02%3A_Polynomial_and_Rational_Functions/2.03%3A_Analysis_of_Graphs_of_Rational_Functions/2.3.04%3A_Sign_Test_for_Rational_Function_GraphsOnce the asymptotes are known you must use the sign testing procedure to see if the function becomes increasingly positive or increasingly negative near the asymptotes. For the purposes of PreCalculus...Once the asymptotes are known you must use the sign testing procedure to see if the function becomes increasingly positive or increasingly negative near the asymptotes. For the purposes of PreCalculus, the testing number should be closer to the vertical asymptote than any other number in the problem. Identify the vertical asymptotes and use the sign testing procedure to roughly sketch the nature of the function near the vertical asymptotes.
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/05%3A_Vector_Analysis/5.02%3A_Vector_Calculations/5.2.03%3A_Cross_ProductsTherefore, the maximum value for the cross product occurs when the two vectors are perpendicular to one another, but when the two vectors are parallel to one another the magnitude of the cross product...Therefore, the maximum value for the cross product occurs when the two vectors are perpendicular to one another, but when the two vectors are parallel to one another the magnitude of the cross product is equal to zero. Since we know the magnitudes of the two vectors and the angle between them, we can use the angle-version of the cross-product equation to determine the magnitude of the cross-product:
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/03%3A_Exponential_and_Logarithmic_Functions/3.05%3A_Exponential_and_Logarithmic_Models/3.5.01%3A_Exponential_ModelsThe population does not continue to increase by the same number of people each year, it rather increases by a percentage of the population at the end of each year, an exponential function. For questio...The population does not continue to increase by the same number of people each year, it rather increases by a percentage of the population at the end of each year, an exponential function. For questions 16-20, calculate the number of years required before the value reaches the specified total, using A f =A i ⋅r t and beginning with A f = final amount, and x (in the exponent) as the number of years.
- https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/01%3A_Analyzing_Functions/1.01%3A_Relations_and_Functions/1.1.03%3A_SymmetryEven functions have the property that when a negative value is substituted for x, it produces the same value as when the positive value is substituted for the x. Odd functions have the property that w...Even functions have the property that when a negative value is substituted for x, it produces the same value as when the positive value is substituted for the x. Odd functions have the property that when a negative x value is substituted into the function, it produces a negative version of the function evaluated at a positive value. An even function is a function with a graph that is symmetric with respect to the y-axis and has the property that f(−x)=f(x).
