Slopes of curves

How can we talk about slopes of curves?

  1. What adjectives come to mind?
  2. How would you visualize these adjectives?
  3. Think about what you encountered in high school. What makes lines special?

Building a common starting point for slopes of curves

  1. A function \(y=f(x)\)
  2. A point on the function \((a, f(a))\)
  3. Introduce a symbol for it
  4. Think about how to compute it
  5. Generalize to any valid starting point \(a\)

One of the key components in the definition of a derivative

Definition 1 (Newton quotient) Let \(h\neq 0\). Let \(f\) be a function with two given points \((a, f(a))\) and \((a+h, f(a+h))\). The fraction \[\frac{f(a+h)-f(a)}{h}\] is called the Newton quotient of \(f\) given two points \((a, f(a))\) and \((a+h, f(a+h))\).

Example 01

Let \(f(x)=mx+b\). Compute the Newton quotient of \(f\).

Example 02

Let \(g(r)=\pi r^2\). Compute the Newton quotient of \(g\).

Example 03

Let \(f(x) = \sqrt{x}\). Compute the Newton quotient of \(f\).

Alternative terms

The Newton quotient is sometimes referred to as:

  • difference quotient: Term used when calculating slopes of curves using a computer
  • average rate of change: Why?

A “wordy” definition of the derivative

The derivative of a function \(f\) at \((a, f(a))\):

  • denoted by \(f^{\prime}(a)\)
  • the resulting Newton quotient given two points \((a, f(a))\) and \((a+h, f(a+h))\) after letting \(h\neq 0\) become closer and closer to 0.

Revisiting our examples

  1. \(f(x)=mx+b\)
  2. \(g(r)=\pi r^2\)
  3. \(f(x) = \sqrt{x}\)

The derivative at any point

  • Since \(f^{\prime}(a)\) is the derivative at a point \((a, f(a))\), we can extend the idea to other points for which the derivative is defined.

  • So when you are given \(f(x)\), we can find the derivative \(f^{\prime}(x)\).

  • Of course, the derivative need not be defined at every point.

  • An example is the function \(f(x)=|x|\). Here calculating \(f^\prime(0)\) will make you feel jarring.

What happens if a function is a bit more complicated?

Let us revisit a function from an earlier video: \[c(t)=\frac{t}{t^2+4}\]

Find \(c^\prime(t)\).

Forming the line tangent to a curve at a point

  1. We are given the function \(f\) and a point \((a, f(a))\).
  2. We also have the derivative \(f^{\prime}(a)\).
  3. How do you form the equation of the line tangent to a curve at a point?

What do we gain from forming the line tangent to a curve at a point?

  1. We become one step closer to creating approximations of complicated functions.
  2. The linear approximation to \(f\) about \(x=a\) is \[P_1(x)=f(a)+f^{\prime}(a)(x-a)\]
  3. We can also use the linear approximation as a way to predict what \(f(x)\) could be using \(P_1(x)\).

Picture time (\(t=1\))

Picture time (\(t=2\))

Picture time (\(t=3\))

Picture time (\(t=4\))

Picture time (\(t=4\)): Zooming in

Picture time (\(t=4\)): Zooming in

Picture time (\(t=4\)): Zooming in

Picture time (\(t=4\)): Zooming in

Looking forward

  • Surely, there has to be shortcuts in calculating the derivative.
  • How do we interpret these derivatives? We actually already started.
  • We may have to dig in a bit into the concept of getting “closer and closer”.
  • What if we have even more complicated functions?
  • Where is the economics?