I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 3.2 : Interpretation of the Derivative
8. What is the equation of the tangent line to \(f\left( x \right) = 3 - 14x\) at \(x = 8\).
Show SolutionWe know that the derivative of a function gives us the slope of the tangent line and so we’ll first need the derivative of this function. We computed this derivative in Problem 2 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.
Note that we did use a different set of letters in the previous problem, but the work is identical. So, from our previous work (with a corresponding change of variables) we know that the derivative is,
\[f'\left( x \right) = - 14\]This tells us that the slope of the tangent line at \(x = 8\) is then : \(m = f'\left( 8 \right) = - 14\). We also know that a point on the tangent line is : \(\left( {8,f\left( 8 \right)} \right) = \left( {8, - 109} \right)\).
The tangent line is then,
\[y = - 109 - 14\left( {x - 8} \right) = 3 - 14x\]Note that, in this case the tangent is the same as the function. This should not be surprising however as the function is a line and so any tangent line (i.e. parallel line) will in fact be the same as the line itself.