- Calculate the slope of a tangent line to a curve and find its equation
- Find the derivative of a function at a given point
- Explain how velocity measures speed over time, and compare average velocity over a period with the exact speed at a specific moment
Tangent Lines
The Main Idea
- Secant Lines and Difference Quotients:
- Slope of a secant line: [latex]m_{\text{sec}} = \frac{f(x) - f(a)}{x - a}[/latex]
- Difference quotient with increment [latex]h[/latex]: [latex]Q = \frac{f(a+h) - f(a)}{h}[/latex]
- Tangent Lines:
- Limit of secant lines as they approach a point
- Slope of tangent line: [latex]m_{\text{tan}} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}[/latex] or [latex]m_{\text{tan}} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}[/latex]
- Local Linearity:
- Near the point of tangency, the function appears linear
- Tangent line provides a good approximation of the function
- Equation of Tangent Line:
- Use point-slope form: [latex]y - y_1 = m(x - x_1)[/latex]
- [latex](x_1, y_1)[/latex] is the point of tangency, [latex]m[/latex] is the slope of the tangent line
Find the equation of the line tangent to the graph of [latex]f(x)=\dfrac{1}{x}[/latex] at [latex]x=2[/latex].
Find the slope of the line tangent to the graph of [latex]f(x)=\sqrt{x}[/latex] at [latex]x=4[/latex].
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You can view the transcript for this segmented clip of “3.1 Defining the Derivative” here (opens in new window).
Find the equation of the line tangent to [latex]f(x) = x^2[/latex] at [latex]x = 3[/latex].
The Derivative of a Function at a Point
The Main Idea
- Definition of the Derivative:
- The derivative is the instantaneous rate of change of a function at a point
- It’s equivalent to the slope of the tangent line at that point
- Notations for the Derivative:
- [latex]f'(a)[/latex] denotes the derivative of [latex]f(x)[/latex] at [latex]x = a[/latex]
- Two Equivalent Definitions:
- [latex]f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}[/latex]
- [latex]f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}[/latex]
- Formal definition of a derivative:
The derivative of a function [latex]f(x)[/latex] at a point [latex]a[/latex] is defined as:
[latex]f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}[/latex]
or equivalently:
[latex]f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}[/latex]
provided these limits exist.
For [latex]f(x)=3x^2-4x+1[/latex], find [latex]f^{\prime}(2)[/latex] by using the first definition.
For [latex]f(x)=x^2+3x+2[/latex], find [latex]f^{\prime}(1)[/latex].
Watch the following video to see the worked solution to this example.
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You can view the transcript for this segmented clip of “3.1 Defining the Derivative” here (opens in new window).
Find [latex]f'(2)[/latex] for [latex]f(x) = 3x^2 - 4x + 1[/latex] using the second definition.
Velocities and Rates of Change
The Main Idea
- Average Velocity:
- For a position function [latex]s(t)[/latex], average velocity over [latex][a,t][/latex] is: [latex]v_{\text{avg}} = \frac{s(t) - s(a)}{t - a}[/latex]
- Instantaneous Velocity:
- Limit of average velocity as time interval approaches zero
- Equivalent to the derivative of the position function: [latex]v(a) = s'(a) = \lim_{t \to a} \frac{s(t) - s(a)}{t - a}[/latex]
- Instantaneous Rate of Change:
- Generalization of instantaneous velocity to any function
- For a function [latex]f(x)[/latex], the instantaneous rate of change at [latex]a[/latex] is [latex]f'(a)[/latex]
- Graphical Interpretation:
- Average velocity: Slope of secant line
- Instantaneous velocity: Slope of tangent line
A rock is dropped from a height of [latex]64[/latex] feet. Its height above ground at time [latex]t[/latex] seconds later is given by [latex]s(t)=-16t^2+64, \, 0\le t\le 2[/latex]. Find its instantaneous velocity [latex]1[/latex] second after it is dropped, using the definition of a derivative.
Watch the following video to see the worked solution to this example.
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You can view the transcript for this segmented clip of “3.1 Defining the Derivative” here (opens in new window).
A toy company can sell [latex]x[/latex] electronic gaming systems at a price of [latex]p=-0.01x+400[/latex] dollars per gaming system. The cost of manufacturing [latex]x[/latex] systems is given by [latex]C(x)=100x+10,000[/latex] dollars. Find the rate of change of profit when [latex]10,000[/latex] games are produced. Should the toy company increase or decrease production?