259 Lecture 5 Spring 2016 Mathematical Functions in Excel

2 Mathematical Functions  Excel has many built-in mathematical functions!  The complete list can be found online here: us/excel-help/math-and- trigonometry-functions-reference- HP aspx?CTT=1http://office.microsoft.com/en- us/excel-help/math-and- trigonometry-functions-reference- HP aspx?CTT=1  Here are some familiar mathematical functions:

3 Common Mathematical Functions  SQRT  ABS  EXP  LN  LOG10  POWER Raises a number to a specified power.  ROUND Rounds a number to a specified number of decimal places.  SIN  COS  TAN  CSC  SEC  COT  PI  RADIANS

4 Trigonometric Functions  In Excel, mathematical functions work as one would expect!  For example, the syntax for the sine function is: SIN(number), where number is the angle in radians for which you want the sine.  Note that if an argument is in degrees, you can use the functions PI or RADIANS to convert the number to radians!

5 Example 1  Make a table for f(x) = sin x, for x in the x-interval [0, 2], in increments of /8.  Plot the graph of y = sin x on the interval [0, 2].  How can the graph be refined to look more like what we are used to seeing (on paper or on a graphing calculator)?

6 Example 1 (cont.)

7 Example 2  Create the function g(x) = tan x, using the sine and cosine functions in Excel.  Compare your created tangent function g(x) to the actual built-in tangent function!  Make a table of tangent function values and plot this function, as we did in Example 1.

8 Example 2 (cont.)

9 Best-Fit Lines Revisited!  Recall that for data points {(x 1,y 1 ), (x 2,y 2 ), …, (x n,y n )}, the best-fit line is defined by y = a+bx, with  Using the SUMPRODUCT function, we can compute best-fit lines more efficiently!

10 The SUMPRODUCT Function  Syntax: SUMPRODUCT(array1,array2,array3,...) where array1, array2, array3,... are 2 to 255 arrays whose components you want to multiply and then add.  Multiplies corresponding components in the given arrays, and returns the sum of those products.  The array arguments must have the same dimensions. If they do not, SUMPRODUCT returns the #VALUE! error value.  SUMPRODUCT treats array entries that are not numeric as if they were zeros.

Example 3  Construct a best-fit line for the toad data, using the SUMPRODUCT function. 11 YearArea(km^2)

12 Example 3 (cont.)

13 Example 4  Another way to find a best-fit line for some data is with the SLOPE and INTERCEPT functions!  Repeat Example 3 with these functions.  To do so, we need to know what these functions do!

14 The SLOPE Function  Syntax: SLOPE(known_y's,known_x's)  known_y's is the dependent set of observations or data.  known_x's is the independent set of observations or data.  Calculates the slope of the best-fit regression line plotted through data points in known_x's and known_y's.  The arguments should be either numbers or names, arrays, or references that contain numbers.  If an array or reference argument contains text, logical values, or empty cells, those values are ignored; however, cells with the value zero are included.  If known_y's and known_x's are empty or have a different number of data points, SLOPE returns the #N/A error value.

15 The INTERCEPT Function  Syntax: INTERCEPT(known_y's,known_x's)  known_y's is the dependent set of observations or data.  known_x's is the independent set of observations or data.  Calculates the y-intercept of the best-fit regression line plotted through data points in known_x's and known_y's.  The arguments should be either numbers or names, arrays, or references that contain numbers.  If an array or reference argument contains text, logical values, or empty cells, those values are ignored; however, cells with the value zero are included.  If known_y's and known_x's are empty or have a different number of data points, INTERCEPT returns the #N/A error value.

16 Example 4 (cont.)

17 A Better Trendline for the Toads Data  Using Excel’s Trendline feature, we can find a function that fits the data better than a linear function!  It turns out that an exponential function does a much better job!

18 Example 5  Using the exponential trendline found by Excel, along with the POWER and EXP function, compare the actual toad data to that found with the exponential trendline y = 9* e x.  Note that Excel 2007 may give y = 9* e 0.077x.

Example 5 (cont.)  A way to fix the “missing digits” in the trendline equation can be found here:  Unfortunately, this may introduce a new problem! 19

20 Rates of Change  Excel is useful for creating function tables to investigate rates of change!  Recall that for a function y = f(x), the average rate of change between points (x 1,f(x 1 )) and (x 2,f(x 2 )) is given by:  The instantaneous rate of change at the point (x 1,f(x 1 )) is found by taking the limit provided this limit exists.

21 Rates of Change (cont.)  If we let x 1 = a and x 2 = a + h, then our definitions become:  Average rate of change of y = f(x) between points (a,f(a)) and (a+h,f(a+h)):  Instantaneous rate of change of y = f(x) at the point (a,f(a)): provided this limit exists.  An idea related to rates of change is that of tangent line.

22 The Tangent Line  The line tangent to the graph of the function y = f(x), at the point (a,f(a)) is the line through the point (a,f(a)), with slope m tan given by provided this limit exists.  Notice that m tan is the instantaneous rate of change of y = f(x) at the point (a,f(a))!

23 The Derivative  Corresponding to x = a in the domain of f(x), for which the graph of y = f(x) has a tangent line at (a,f(a)), is exactly one slope.  Thus, we can define a function that specifies the slope of the tangent line to y = f(x) when x = a.  The derivative of the function y = f(x) at x = a is the number f’(a), given by provided this limit exists.  The derivative f’(a) gives the instantaneous rate of change of f with respect to x when x = a.

24 Example 6  If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torriceli’s Law gives the volume V of the water remaining in the tank after t minutes as  Find the average rate at which the water is draining out of the tank between times t = 10 min and t = 20 min t = 10 min and t = 15 min t = 10 min and t = 11 min t = 10 min and t = 10.1 min t = 10 min and t = min t = 10 min and t = min t = 10 min and t = min t = 10 min and t = min  Estimate the instantaneous rate at which water is flowing out of the tank at t = 10 min.  (If time) Graph y = V(t) from Example 6, along with the tangent line y = L(t) at t = 10 on the same xy-coordinate axes.

25 Example 6 (cont.)

26 Engineering Functions  In addition to the standard mathematical and trigonometric functions, Excel has several built-in functions that are useful in applied mathematics areas, including engineering!  These functions may need to be added in to the set of available functions.

Engineering Functions (cont.)  To see if the Engineering Functions are included, look in the Function Library group in the Formulas tab.  You may have to click on the More Tools drop-down menu.  If Engineering Functions is not listed, you will have to add them in, via Add-Ins. 27

Loading Excel Add-Ins  Click the Microsoft Office Button, and then click Excel Options.  Click the Add-Ins category.  In the Manage box, click Excel Add-ins, and then click Go.  To load an Excel add-in, do the following: In the Add-Ins available box, select the check box next to the add-in that you want to load, and then click OK. Tip If the add- in that you want to use is not listed in the Add-Ins available box, click Browse, and then locate the add-in. Add-ins that are not available on your computer can be downloaded from Downloads on Office Online.Downloads If the add-in is not currently installed on your computer, click Yes to install it. Tip Follow the setup instructions as needed.  To unload an Excel add-in, do the following: In the Add-Ins available box, clear the check box next to the add-in that you want to unload, and then click OK. To remove the add-in from the Ribbon, restart Excel. 28

29 Engineering Functions (cont.)  Examples of the functions available include:  BIN2DEC, which converts a binary number (base 2) to a decimal number.  HEX2DEC, which converts a hexadecimal number (base 16) to a decimal number.  CONVERT, which converts a number in one measurement system to another.  COMPLEX, which turns a pair of real numbers into a complex number.  IMPRODUCT, which multiplies complex numbers.

30 Example 7  Try each of the following commands:  BIN2DEC( )  HEX2DEC(“FF”) - HEX2DEC(“F8”)  CONVERT(50, “mi”, “km”)  COMPLEX(2,3)  IMPRODUCT(“2+3i”, “1-i”)

31 References  James Stewart, Calculus (Early Transcendentals), 5 th edition  Microsoft online help: us/excel-help/math-and- trigonometry-functions-reference- HP aspx?CTT=1 us/excel-help/math-and- trigonometry-functions-reference- HP aspx?CTT=1