Defining the Derivative: Fresh Take

  • 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].

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].

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.

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?