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Section Tangent Lines

We are going to define tangent line from a physical point-of-view rather than give a rigorous mathematical definition. Sketch any function, \(f\text{,}\) and pick a point on it. Label your point \((a,f(a)).\) Place your pencil on the graph at a point to the left of \((a,f(a)).\) Imagine that your pencil is a car, the graph is a road, and you are driving toward the point \((a,f(a))\) from the left. Suppose that there is a perfectly frictionless patch of ice at the point \((a,f(a))\) and keep drawing in the direction the car would travel as it slides off the road. Now do the same from the right. If the two lines you drew form a line, then you have drawn the tangent line to \(f\) at the point \((a,f(a))\). If they don't form a line, then \(f\) does not have a tangent line at this point.

Definition 1.63.

A secant line for the graph \(f\) is a line passing through two points on the graph of \(f.\)

Problem 1.64.

Let \(f(x) = x^2+1\) and graph \(f.\) Graph and compute the slope of the secant lines passing through:

  1. \((2,f(2))\) and \((4,f(4))\)

  2. \((2,f(2))\) and \((3,f(3))\)

  3. \((2,f(2))\) and \((2.5,f(2.5))\)

  4. \((2,f(2))\) and \((2.1,f(2.1))\)

What is the slope of the line tangent to \(f\) at the point \((2,5)\text{?}\)

Problem 1.65.

Write down a limit that represents the work you did in the last problem.

Problem 1.66.

Find the slope of the tangent line to the curve \(f(x)= x^3 + 2\) at the point \((3, 29)\text{.}\)

Problem 1.67.

Find the equation of the tangent line in Problem 1.66.

Problem 1.68.

Assume \(a \in \re.\) Find the equation of the line tangent to the graph of \(f(x) = x^3 + 2\) at the point \((a,f(a)).\)

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