TL4 Tangent Lines - MTH 121 Spring 2026

Summary So Far¶

Let’s recap what we have seen so far:

TL1: How to write equations for general lines.
TL2: Estimating the slope of a curve by drawing the tangent line.
TL3: Calculating the derivative of a curve to get the slope formula.

Now we’re going to put all these pieces together to be able to write down the equation of a tangent line. We start with our usual line equation: the point-slope formula. (Remember, this equation works for any line, not just tangent lines!)

Core Concepts¶

Point-Slope Formula

(y-y_c)=m(x-x_c)

In order to use this formula, we need 3 ingredients:

$x$ -coordinate: $x_c$
$y$ -coordinate: $y_c$
slope: $m$

Tangent Line Formula

y-f(x_c)=f'(x_c)(x-x_c)

In order to use this formula, we need 3 ingredients:

$x_c$ is usually given
$y_c=f(x_c)$
$m=f'(x_c)$

We see that this is where Calculus comes into play. We use the derivative $f'(x)$ to find the slope of the tangent line (slope of our curve)

Primary Examples¶

Homework¶

Write out each question as well as your solution on the homework page template.

Make sure you leave a good amount of blank space between each question.
Write down the question as well as your solution.