Slopes of Tangent Lines and Derivatives: Calculus Study Guide

Slopes of Tangent Lines

Secant and Tangent Lines: Rate of Change

Understanding the difference between secant and tangent lines is fundamental in calculus, as they relate to average and instantaneous rates of change, respectively.

• Secant Line: Intersects a curve at two points. The slope represents the average rate of change between those points.

• Tangent Line: Touches the curve at exactly one point. The slope represents the instantaneous rate of change at that point.

• Derivative: The slope of the tangent line at a point is called the derivative of the function at that point.

Formulas

• Average Rate of Change (Secant): $m_{sec} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Instantaneous Rate of Change (Tangent): $m_{tan} = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$

Example

• Given $f(x) = x^2$, the slope of the tangent line at $x = 1$ is: $m_{tan} = \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1} (x+1) = 2$

Equations of Tangent Lines

Finding the Equation of a Tangent Line

The equation of a tangent line to a curve at a given point can be found using the point-slope form and the derivative.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $m$ is the slope at $x_1$.

• To find $m$, use the definition of the derivative: $m = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$

Step-by-Step Procedure

1. Plug $x$-value ($c$) into $f(x)$ to get $y$-value ($f(c)$).

2. Plug $f(x)$, $c$, and $f(c)$ into the derivative formula.

3. Evaluate the limit to find $m$.

4. Plug $m$ and the point into the point-slope equation.

5. Solve for $y$ if needed.

Example

• Find the equation of the tangent line to $f(x) = 3x^2 - 4$ at $x = 2$: $m_{tan} = \lim_{x \to 2} \frac{3x^2 - 4 - (3(2)^2 - 4)}{x - 2}$ $= \lim_{x \to 2} \frac{3x^2 - 4 - 8}{x - 2}$ $= \lim_{x \to 2} \frac{3x^2 - 12}{x - 2}$ $= \lim_{x \to 2} \frac{3(x^2 - 4)}{x - 2}$ $= \lim_{x \to 2} \frac{3(x-2)(x+2)}{x-2}$ $= \lim_{x \to 2} 3(x+2) = 12$ Equation: $y - 8 = 12(x - 2)$

Derivatives

Definition and Notation

The derivative of a function at a point measures the instantaneous rate of change, or the slope of the tangent line at that point.

• Derivative Notations: $f'(x)$, $\frac{dy}{dx}$

• Definition of the Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Example

• Find the derivative of $f(x) = x^2$: $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$ $= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$ $= \lim_{h \to 0} \frac{2xh + h^2}{h}$ $= \lim_{h \to 0} (2x + h) = 2x$

Graphing the Derivative

Relationship Between $f(x)$ and $f'(x)$

The graph of the derivative function $f'(x)$ provides information about the slope of the original function $f(x)$ at each point.

• Where $f(x)$ is increasing, $f'(x) > 0$.

• Where $f(x)$ is decreasing, $f'(x) < 0$.

• Where $f(x)$ has a horizontal tangent (local max/min), $f'(x) = 0$.

• Discontinuities or sharp corners in $f(x)$ may cause $f'(x)$ to be undefined or have jumps.

Example Table

Interval	$f(x)$ (slope)
$(-\infty, -1)$	positive
$(-1, 1)$	negative
$(1, \infty)$	positive

Sketching $f'(x)$

• For each $x$-value or interval, determine if the slope of $f(x)$ is positive, negative, or zero, and sketch $f'(x)$ accordingly.

Graphing the Derivative – Special Cases

Discontinuities and Jumps

If there is a discontinuity or sharp corner on the graph of $f(x)$, then $f'(x)$ does not exist (DNE) and has a jump at that point.

• The slope of any line is constant, so the derivative of a linear function is a constant.

Example Table

Interval	$f(x)$ (slope)
$(-\infty, -1)$	positive
$(-1, 1)$	zero
$(1, 0)$	negative
$(0, \infty)$	positive

Example

• Given a piecewise function with a jump at $x = 2$, the derivative $f'(x)$ would have a discontinuity at $x = 2$.

Summary Table: Key Concepts

Concept	Definition	Formula
Secant Line	Average rate of change	$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$
Tangent Line	Instantaneous rate of change	$\lim_{x \to c} \frac{f(x) - f(c)}{x - c}$
Derivative	Slope of tangent line	$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Practice Problems

• Calculate the slope of the tangent line for $f(x) = 4x^2 - 1$ at $x = -3$.

• Find the equation of the tangent line to $f(x) = 3(x^2 - 1)$ at $x = 1$.

• Use the definition of the derivative to find $g'(x)$ for $g(x) = x^3$ at $x = -1$.

• Sketch the derivative function $f'(x)$ given the graph of $f(x)$.

Additional info: These notes cover foundational calculus concepts including the definition and calculation of derivatives, the geometric interpretation of tangent and secant lines, and graphical analysis of derivatives. Practice problems and tables are included to reinforce understanding and application.