1.9.1: Resources and Key Concepts

Resources

Videos

• Secant line
  • Limits - Secant and Tangent Lines - Secant Lines and Average Rates of Change: Definitions & Examples
  • Limits - Secant and Tangent Lines - Tangent Lines and Instantaneous Rates of Change: Definition & Examples
• Tangent line
  • Derivatives and Rates of Change - Recalling Tangent Lines
• Derivative at a point
  • Derivatives and Rates of Change - The Definition of the Derivative at a Point
  • Derivatives and Rates of Change - Finding the Derivative of a Function at a Point
  • Derivatives and Rates of Change - Finding the Derivative of a Function at an Arbitrary Point
  • Derivatives and Rates of Change - Interpreting Limits as Derivatives
• Average velocity
  • Limits - Secant and Tangent Lines - Average and Instantaneous Velocities
• Instantaneous velocity
  • Derivatives and Rates of Change - Rates of Change and Instantaneous Velocity
• Equation of tangent line
  • Derivatives and Rates of Change - Finding Equations of Tangent Lines

Key Concepts

Definitions

• Difference Quotient:
  • Given a function \(f\) defined on an interval \(I\) containing \(a\), if \(x \neq a\) is in \(I\), then \(Q = \frac{f(x)-f(a)}{x-a}\) is a difference quotient.
  • If \(h \neq 0\) is chosen so that \(a+h\) is in \(I\), then \(Q = \frac{f(a+h)-f(a)}{h}\) is a difference quotient with increment \(h\).
• Tangent Line: Let \(f(x)\) be a function defined in an open interval containing \(a\). The tangent line to \(f(x)\) at \(a\) is the line passing through the point \((a, f(a))\) having slope:
  • \(m_{tan} = \displaystyle \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\)
  • or equivalently, \(m_{tan} = \displaystyle \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\)
  provided these limits exist.
• Derivative: Let \(f(x)\) be a function defined in an open interval containing \(a\). The derivative of the function \(f(x)\) at \(a\), denoted by \(f'(a)\), is defined by:
  • \(f'(a) = \displaystyle \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\)
  • or alternatively, \(f'(a) = \displaystyle \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\)
  provided this limit exists.
• Differentiation: The process of finding a derivative.
• Average Velocity: If \(s(t)\) is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval \([a,t]\) (or \([t,a]\)) is given by the difference quotient \(v_{avg} = \frac{s(t)-s(a)}{t-a}\).
• Instantaneous Velocity: As the values of \(t\) approach \(a\), the values of \(v_{avg}\) approach the value we call the instantaneous velocity at \(a\). Denoted \(v(a)\), it is given by \(v(a) = s'(a) = \displaystyle \lim_{t \to a} \frac{s(t)-s(a)}{t-a}\).
• Instantaneous Rate of Change: The instantaneous rate of change of a function \(f(x)\) at a value \(a\) is its derivative \(f'(a)\).