Given a role, you have the right to easily uncover the slope of a tangent line using Microsoft Excel to execute the dirty work. That is to say, you have the right to input your x-value, produce a pair of formulas, and also have Excel calculate the secant value of the tangent slope. This is a fantastic tool for Stewart Calculus sections 2.1 and also 2.2.
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Note that you deserve to basically enter the exact very same attributes into Google Sheets as well as Excel, so this is a good tool for either choice. So if you have a Windows or Mac machine via Excel mounted, usage that. If you"re running a Linux distro or Chrome OS, or would certainly fairly usage a complimentary tool, usage Google Sheets.
For this example, we"ll usage the adhering to function:
To analyze, we desire the to discover the slope of the tangent line of the function 2+SQRT(x) where x=1.
Image using Robert Talbert
Tip 1: Find the Y-Coordinate
Using the formula, it"s fairly straightforward to uncover the y-coordinate based upon the x-worth. For example, for x=1, the y-coordinate would certainly be F(1)=2+SQRT(1), leading to a worth of 3.
Image by means of Robert Talbert
Tip 2: Secant Lines
Now we"ll have to usage points both before and after the x-worth that approach that value, however never reach it (purple line below). Plotting these points on the graph will certainly produce the secant line for the formula.
Image by means of Robert Talbert
Now it"s time to begin plotting. We"ll begin with hand also calculations prior to relocating on to making use of Excel (or Sheets) to automate the procedure.
We"ll should plot points for the x-worth, the y-value, and also the secant slope.
Image via Robert Talbert
As you deserve to see over, through x=2, the y-value is 3.4142 (utilizing the formula). The secant slope is calculated by taking the difference in the y-worth and also dividing it by the difference in the x-value, which is 0.4142.
Step 3: Automate Y-Value via Excel
Recontact that we desire the x-value to approach 1 without ever getting to it. So, we can usage 2, 1.5, 1.1, 1.01, and also 1.001. You"ll also want to perform this coming the other method, so we have the right to use 0, 0.5, 0.9, 0.99, and also 0.999.
Start by labeling the columns, then enter your x-values. We"ll begin through 2 as it ideologies 1.
Now enter the formula to calculate the y-worth. In the second column, input "=2 + sqrt(A2)" wbelow A2 is package wright here the x-worth is stored (you deserve to enter "A2" manually, or ssuggest click package that contains the value), then hit enter.
Now click the box that you simply entered the formula into, then drag from the bottom-appropriate of that box dvery own to the last x-worth to insert the formula right into each one—you"ll alert the y-values pop up.
Step 4: Automate Secant Slope via Excel
Now in the 3rd column, let"s enter the formula to calculate the secant slope. The formula below is "=(B2 - 3)/(A2 - 1)" (again, you have the right to ssuggest click the boxes that hold the value).
Just as prior to, drag dvery own from the bottom-appropriate to use the formula for the rest of the x- and y-worths.
Tip 5: Calculate for X-Values from 0 Approaching 1
Now simply rinse-and-repeat, making use of x-worths going the other way. Apply the exact same drag method for these.
Tright here you have actually it, the secant slopes for this formula when x=1. For a better information, check out the video below.
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