259 Lecture 5 Spring 2015

Mathematical Functions in Excel

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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:

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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

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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!

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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)?

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Example 1 (cont.)

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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.

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Example 2 (cont.)

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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!

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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.

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Year Area(km^2)

1939 32800

1944 55800

1949 73600

1954 138000

1959 202000

1964 257000

1969 301000

1974 584000

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Example 3 (cont.)

13568.1

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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!

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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.

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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.

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Example 4 (cont.)

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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!

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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!

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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.

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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.

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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))!

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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.

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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.

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Example 6 (cont.)

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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.

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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.

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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.

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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”)

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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