Problem #6

Using Winplot Animation on Secant Lines vs. Tangent Lines

Problem Description

Using the function , prove the slope of the secant line between x=1 and x=1+h gets closer to the slope of the tangent line as h approaches 0.

Let’s start with

graphing the equation:

𝑦=14 𝑥

2

x

y

Next, we shall find the tangent line:• . x

• m at point (1, )

• To find the tangent line, you must first take the derivative of the equation

• Now lets find the slope of the tangent line at point at x=1.

• Now lets graph the equation of the tangent line using the slope formula

• Find point of intersection of x=1 using the original equation.

• Multiply the (x-1) with and add to both sides.

• This gives us the equation of the tangent line.

Here is the graphs of:

and the tangent line

x

y

Now, we shall find the secant line:

• Use the Secant Formula:Note: Remember F(x) is another term for y.

• Enter into the formula above

• Remember x=1.• Simplify• Multiply • Simplify.• Factor out an h.• Simplify • h

Equation of the Secant Line

• Use the slope formula to graph the equation of the secant line.

• Remember the known point is (1,).

Secant Lines

• Let us begin by letting h=5

• Equation is:

• Simplify:

x

y

Secant Lines

• Let us begin by letting h=4

• Equation is:

• Simplify:

x

y

Secant Lines

• Let us begin by letting h=3

• Equation is:

• Simplify:• 1

x

y

Secant Lines

• Let us begin by letting h=2

• Equation is:

• Simplify:

x

y

Secant Lines

• Let us begin by letting h=1

• Equation is:

• Simplify:

x

y

Secant Lines

• Let us begin by letting h=

• Equation is:

• Simplify:

x

y

Summary

• Using the previous slides we proved that• Has a tangent line at: • And the secant line between x=1 and x=1+h gets

closer to the equation of the tangent line as h gets closer to 0.

h=5h=4h=2 h=