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159 Lecture 5 Fall 2009

159 Lecture 5 Fall 2009. 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/CH100645361033.aspx Here are some familiar mathematical functions:. SQRT

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159 Lecture 5 Fall 2009

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  1. 159 Lecture 5 Fall 2009 Mathematical Functions in Excel

  2. Mathematical Functions • Excel has many built-in mathematical functions! • The complete list can be found online here: http://office.microsoft.com/en-us/excel/CH100645361033.aspx • Here are some familiar mathematical functions:

  3. 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 Common Mathematical Functions

  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 RADIANSto 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 {(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!

  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.

  11. Example 3 • Construct a best-fit line for the toad data, using the SUMPRODUCT function.

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

  19. 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!

  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 (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.

  21. 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.

  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 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))!

  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 = 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.

  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.

  27. 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.

  28. 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.

  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(111000101) • HEX2DEC(“FF”) - HEX2DEC(“F8”) • CONVERT(50, “mi”, “km”) • COMPLEX(2,3) • IMPRODUCT(“2+3i”, “1-i”)

  31. References • James Stewart, Calculus (Early Transcendentals), 5th edition • Microsoft online help: http://office.microsoft.com/en-us/excel/CH100645361033.aspx

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