Unit 1 Summary

Functions

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Recognize a linear function from a formula, table of data or a graph.
Fit a linear function to data
- The line with slope \(m\) and vertical intercept \(b\) is \(y=mx+b\)
- The line with slope \(m\) containing point \((a,b)\) is \((y-b) = m(x-a)\)
- The line containing points \((a_1,b_1)\) and \((a_2,b_2)\) is \(\displaystyle{y-b_1 = \frac{b_2-b_1}{a_2-a_1} (x-a_1)}\).

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Calculate the average rate of change (AROC) of function \(y=f(x)\) on the interval \([a,b]\)
Determine where a function is increasing and where it is decreasing
Determine where a function is concave up and where it is concave down

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Perform calculations using exponentials and logarithms
Explain the difference between continuous exponential growth (instantaneous growth) and discrete exponential growth (growth at specific intervals)
Model continuous exponential growth using \(P(t) = P_0 e^{rt}\)
Model discrete exponential growth using \(P(t) = P_0 (1+r)^t\)
Find the exponential function that fits a set of data
Given a half life or doubling time, find the corresponding rate of growth \(r\).
Given an exponential function, find its half life or its doubling time

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Describe proportional and inversely proportional relationships
Recognize power functions by creating a log-log plot.
Enter a table of data into desmos
Use desmos to fit a power function \(y=kx^p\) to data using a command like \(\ln (y_1) \sim c + p \ln(x_1)\) or \(\ln (y_1) \sim \ln(k) + p \ln(x_1)\)

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Describe periodic phenomena using words, equations and graphs.
Explain the effect of changing the constants \(A,C,\omega,\phi\) in the functions \(f(t) = A\sin(\omega(t+\phi)) +C\) and \(g(t) = A\cos(\omega(t+\phi)) +C\)
Model periodic data (graph or table) by choosing sine or cosine and estimating the parameters \(A,C,\omega,\phi\).
- \(A = \frac{\max-\min}{2}\) is the amplitude
- \(C = \frac{\max+\min}{2}\) the center
- \(\omega = \frac{2\pi}{\mbox{period}}\) is the frequency
- \(\phi\) is the horizontal shift

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Work with functions when the output depends on multiple inputs
Interpret functions of the form \(z=f(x,y)\) represented as formulas, tables of data, or contour plots
Create and interpret a cross section of a function of two variables
Use RStudio to create a contour plot of \(z=f(x,y)\) and interpret what I see.