Mathematical Functions in Excel 259 Lecture 5 Spring 2017 Mathematical Functions in Excel

Mathematical Functions Excel has many built-in mathematical functions! The complete list can be found online here: http://office.microsoft.com/en-us/excel-help/math-and-trigonometry-functions-reference-HP010079189.aspx?CTT=1 Here are some familiar mathematical functions:

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

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!

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)?

Example 1 (cont.)

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.

Example 2 (cont.)

Best-Fit Lines Revisited! Recall that for data points {(x1,y1), (x2,y2), …, (xn,yn)}, the best-fit line is defined by y = a+bx, with Using the SUMPRODUCT function, we can compute best-fit lines more efficiently!

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. Year Area(km^2) 1939 32800 1944 55800 1949 73600 1954 138000 1959 202000 1964 257000 1969 301000 1974 584000

Example 3 (cont.) 13568.1

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!

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.

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.

Example 4 (cont.)

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!

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*10-62e0.0779x. Note that Excel 2007 may give y = 9*10-62e0.077x.

Example 5 (cont.) A way to fix the “missing digits” in the trendline equation can be found here: http://support.microsoft.com/kb/282135 Unfortunately, this may introduce a new problem!

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 (x1,f(x1)) and (x2,f(x2)) is given by: The instantaneous rate of change at the point (x1,f(x1)) is found by taking the limit provided this limit exists.

Rates of Change (cont.) If we let x1 = a and x2 = 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.

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 mtan given by provided this limit exists. Notice that mtan is the instantaneous rate of change of y = f(x) at the point (a,f(a))!

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.

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 = 10.01 min t = 10 min and t = 10.001 min t = 10 min and t = 10.0001 min t = 10 min and t = 10.000001 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.

Example 6 (cont.)

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.

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. 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.

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.

Example 7 Try each of the following commands: BIN2DEC(111000101) HEX2DEC(“FF”) - HEX2DEC(“F8”) CONVERT(50, “mi”, “km”) COMPLEX(2,3) IMPRODUCT(“2+3i”, “1-i”)

References James Stewart, Calculus (Early Transcendentals), 5th edition Microsoft online help: http://office.microsoft.com/en-us/excel-help/math-and-trigonometry-functions-reference-HP010079189.aspx?CTT=1