Tangent Lines to Graphs of Functions

When we defined the derivative, we talked about the slope of two points that are very close to one another. In fact, the derivative is defined as:

$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{f(x+h)-f(x)}{(x+h)-x}$

This is the slope of the line created by the two points $x+h$ and $x$ . As $h$ tends to 0, the two points are closer and closer to one another so that the line connecting them becomes a tangent line.

Typically, for a line we are required to have two points. To find a line tangent to a function at a point $x_0$ , all we need is the point, the formula for $f(x)$ and the first derivative.

Table of Contents

Finding the Tangent Line

Given a function $f$ , we know that the first derivative $f'(x)$ and the tangent line are connected. In reality, given any point $x_0$ in the domain of $f$ , the value: $f'(x_0)$ is the slope of the line tangent to the graph of $f$ at the point $x_0$ .

More formally:

Definition (The Tangent Line): Let $f(x)$ be a function that is differentiable at a point $(x_0,y_0)$ . The equation of the line tangent to the graph of $f(x)$ at the point $x_0$ is:

$y - \underbrace{y_0}_{f(x_0)}= f'(x_0) \cdot (x - x_0)$

Geometric Meaning: It is the linear approximation of the curve at that specific location, matching the direction of the graph so that it looks identical to the curve when zoomed in tightly.
Linearization Form: Can be rearranged into function format as $y= f(x_0) + f'(x_0)(x - x_0)$ .

That is, given $x_0$ and $f(x)$ we follow these steps to find the tangent line:

Determine $f'(x)$ .
Calculate $f'(x_0)$ and $f(x_0)$ .
Replace in the equation above.

Example: Find the equation of the tangent line to the function $f(x) = 2x^2 - 3x + 1$ at the point where $x_0 = 2$ .

Example: Find the equation of the tangent line to the function $f(x) = 4\ln(x) + x$ at the point where $x_0 = 1$ .

Problem Solving with Tangent Lines

Using the formula above, we can be given a similar problem disguised differently. For the exercises below, you should remember the following two facts:

Two lines are parallel if they have the same slope.
Two lines are parallel if the product of their slopes is $-1$ , i.e.: $m_1\cdot m_2=-1$ .

Example: Consider the quadratic function $f(x) = 3x^2 + bx - 5$ , where $b$ is an unknown constant parameter. Given that the tangent line to the graph of $f(x)$ at the point where $x_0 = 2$ is parallel to the line $y = 14x + 3$ , find the value of $b$ .

Example: Consider the quadratic function $f(x) = ax^2 - 4x + 7$ , where $a$ is an unknown constant parameter. Given that the tangent line to the graph of $f(x)$ at the point where $x_0 = 1$ is perpendicular to the line $y = \frac{1}{2}x - 9$ , find the value of $a$ .

Similarly, we can be given the same hint in a different way:

Example: Consider the function $f(x) = x^3 + kx^2 - 12x + 5$ , where $k$ is an unknown constant parameter. Given that the tangent line to the graph of $f(x)$ at the point where $x_0 = 2$ is parallel to the $x$ -axis, find the value of $k$ .

Each parameter missing is one piece of information that we need to be given to find it. In the next example, two parameters will be left to calculate, so we will be given two pieces of information and we can create a system of equations from this.

Example: Consider the cubic function $f(x) = ax^3 + bx^2 - 7x + 4$ , where $a$ and $b$ are unknown constant parameters. Given that the curve passes through the point $(1, 2)$ and has a stationary point at this exact location, determine the values of $a$ and $b$ .

Exercises

For each of the following functions, find the equation of the tangent line at the given point $x_0$ . State your final answer in slope-intercept form ( $y = mx + n$ ).

$f(x) = 3x^2 - 5x + 4 \quad \text{at} \quad x_0 = 1$

$f(x) = 2x^3 - 3x^2 \quad \text{at} \quad x_0 = 2$

$f(x) = e^x + 2x \quad \text{at} \quad x_0 = 0$

$f(x) = 3\ln(x) - x^2 \quad \text{at} \quad x_0 = 1$

$f(x) = \frac{8}{x} \quad \text{at} \quad x_0 = 4$

For each of the following, use your knowledge of derivatives, tangent lines, and parallel or perpendicular linear to calculate the value of the unknown parameter ( $a$ , $b$ , or $k$ ).

Consider $f(x) = 2x^2 + bx - 9$ . Find the value of $b$ if the tangent line at $x_0 = 3$ is parallel to the line $y = 16x - 5$ .

$f(x) = ax^2 + 8x + 3$ . Find the value of $a$ if the tangent line at $x_0 = -2$ is parallel to the line $y = -4x + 11$ .

$f(x) = x^3 + kx^2 - 4x + 1$ . Find the value of $k$ if the tangent line at $x_0 = 2$ is perpendicular to the line $y = -\frac{1}{16}x + 8$ .

$f(x) = ax^3 - 5x^2 + 7$ . Find the value of $a$ if the tangent line at $x_0 = 1$ is perpendicular to the line $y = \frac{1}{4}x - 2$ .

$f(x) = k\ln(x) + 3x^2$ for $x > 0$ . Find the value of $k$ if the tangent line at $x_0 = 1$ is parallel to the line $y = 10x - 7$ .

$f(x) = e^{2x} + kx$ . Find the value of $k$ if the tangent line at $x_0 = 0$ is perpendicular to the line $y = -\frac{1}{5}x + 13$ .

Consider the cubic function $f(x) = 2x^3 - 9x^2 + kx - 11$ , where $k$ is an unknown constant parameter. Given that the tangent line to the graph of $f(x)$ at the point where $x_0 = 1$ is parallel to the $x$ -axis, determine the required value of $k$ and state the coordinates of this stationary point.

Consider the cubic function $f(x) = ax^3 + bx^2 + 5x - 3$ , where $a$ and $b$ are unknown constant parameters. Given that the curve passes through the coordinate point $(2, 7)$ and possesses a stationary point at this exact location, construct a system of simultaneous linear equations to determine the precise values of $a$ and $b$ .