Derivatives Study Notes
Chapter 3 | Derivatives
In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter so that we can then explore applications of these techniques.
3.1 | Defining the Derivative
Learning Objectives
Recognize the meaning of the tangent to a curve at a point.
Calculate the slope of a tangent line.
Identify the derivative as the limit of a difference quotient.
Calculate the derivative of a given function at a point.
Describe the velocity as a rate of change.
Explain the difference between average velocity and instantaneous velocity.
Estimate the derivative from a table of values.
Introduction to Calculus
Establishes the foundation for the study of calculus, which includes computing derivatives and integrals.
Historical Context: Calculus was developed independently by:
Isaac Newton (1643-1727, England)
Gottfried Leibniz (1646-1716, Germany)
Both mathematicians contributed significantly, with Newton likely arriving at the concepts first, but Leibniz contributing the notation used today.
Tangent Lines
Secant and Tangent Lines:
Slope of a secant line helps in estimating the rate of change of a function.
Slope Calculation:
For a point \( (a, f(a)) \$ and another point \$ (x, f(x)) \$ near a,
Slope of the secant line \( rac{f(x) - f(a)}{x - a} \$.
Difference Quotient for Secant Line:
Slope can also be found using limit as \$ h \$ approaches 0:
\( rac{f(a + h) - f(a)}{(a + h) - a} = rac{f(a + h) - f(a)}{h} \$.
Definitions
Difference Quotient:
Defined for a function \$ f \$ at point \$ a \$ in an interval I containing \$ a \$ as follows:
\$ Q = rac{f(x) - f(a)}{x - a} \$
Alternative: If \$ h \$ is chosen so that \$ a + h \$ is in I, a different quotient is:
\$ Q = rac{f(a + h) - f(a)}{h} \$.
Relationship Between Secant and Tangent Lines
As the values of \$ x \$ approach \$ a \$ and as \$ h \$ approaches 0, the secant lines approximate the tangent line.
Tangent Line Slope Definition:
The slope of the tangent line at \$ a \$ is given by two limits:
\$ ext{Slope} = ext{lim}_{x o a} rac{f(x) - f(a)}{x - a} \$ (Equation 3.3)
\$ ext{Slope} = ext{lim}_{h o 0} rac{f(a + h) - f(a)}{h} \$ (Equation 3.4)
Example: Finding Tangent Lines
Example 3.1: Finding the equation of the tangent line to \$ f(x) = x^2 \$ at \$ x = 3 \$.
Using Equation 3.3:
Slope is \$ ext{lim}_{x o 3} rac{x^2 - 9}{x - 3} \$
Substitute to find slope: 6
Point on tangent line is (3, 9):
Tangent line equation is \$ y - 9 = 6(x - 3) \$ or simplified to \$ y = 6x - 9 \$.
Alternative Example: Slope Calculation with Equation 3.4
Example 3.2: Use Equation 3.4 for the same tangent line:
Slope is \$ ext{lim}_{h o 0} rac{(3 + h)^2 - 9}{h} \$
Evaluate limit: Slope is also 6.
Finding the Tangent Line for Other Functions
Example 3.3: For \$ f(x) = rac{1}{x} \$ at \$ x = 2 \$:
Slope calculation shows tangent slope is -1, yielding tangent line equation \$ y = -x + 1 \$.
Finding Derivative at a Point:
Definition of derivative: \$ f'(a) = ext{lim}{x o a} rac{f(x) - f(a)}{x - a} \$ (Equation 3.5) or alternatively, \$ f'(a) = ext{lim}{h o 0} rac{f(a + h) - f(a)}{h} \$ (Equation 3.6).
Applications: Velocity and Rates of Change
Average velocity for an object’s position over a time interval \$ [a, t] \$ is:
\$ V_{ave} = rac{s(t) - s(a)}{t - a} \$ (Equation 3.7)
Instantaneous velocity is the slope of the tangent line:
\$ v(a) = s'(a) = ext{lim}_{t o a} rac{s(t) - s(a)}{t - a} \$ (Equation 3.8).
Example of Estimating Velocity
Example 3.7: For a lead weight spring oscillation given by \$ s(t) = ext{sin}(t) \$:
Estimating \$ v(0) \$ using average velocity table produced close to 0 and checking with the derivative gives \$ v(0) = 1 \$.
Summary of Key Definitions
Instantaneous rate of change at a value a: \$ f'(a) \$ is defined and computed using limits.
Examples on Rate of Change
Example 3.8: Acceleration related to vehicular speed data analyses.
Observations on MPH to FT/s to compute average acceleration yield insights about the vehicle's speeding characteristics.
Examples 3.9 and 3.10: Rate of change for temperature modeling in a house as well as profit analysis for a toy company, leading to conclusions on rates of production changes.
Exercises
Exercises generally involve finding secant slopes, estimating tangents, and exploring velocity calculations based on functions given and evaluating using previously defined equations.
These exercises are designed to enhance understanding of derivatives, tangent slopes, and how they apply in practical scenarios including physical motion and economic aspects.