Problem #6

1. Problem #6 Using Winplot Animation on Secant Lines vs. Tangent Lines

2. 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.

3. Let’s start with graphing the equation:

4. Next, we shall find the tangent line: • 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. • . x • mat point (1,)

5. Here is the graphs of:and the tangent line

6. Now, we shall find the secant line: • h • 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

7. Equation of the Secant Line • Use the slope formula to graph the equation of the secant line. • Remember the known point is (1,).

8. Secant Lines • Let us begin by letting h=5 • Equation is: • Simplify:

9. Secant Lines • Let us begin by letting h=4 • Equation is: • Simplify:

10. Secant Lines • Let us begin by letting h=3 • Equation is: • Simplify: • 1

11. Secant Lines • Let us begin by letting h=2 • Equation is: • Simplify:

12. Secant Lines • Let us begin by letting h=1 • Equation is: • Simplify:

13. Secant Lines • Let us begin by letting h= • Equation is: • Simplify:

14. 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=5 h=4 h=2 h=