Visualization of Derivative Part 2
The app below shows the function f(x)=x^2, with point F(2,4) on it. Play around with the app, moving around point M, and seeing what the buttons do. Then, answer the questions in your notebook.
To make the next part easier, we will let the horizontal distance between F and M be h.
Question 1: Since F is at (2,4), and h is the horizontal distance between F and M, what is the x-coordinate of M?
Question 2: M is on the given function, so what is the y-coordinate of M?
Question 3: We want to find the slope of chord FM. Using question 1 and 2, write a formula for the slope of FM, in terms of h.
Question 4: Write the following in your notebook, and fill in the blanks:
"The slope chord of FM will approach the slope of the tangent at F when:
M approaches: _______________
This happens, when h approaches: _____________"
Question 5: Push "Show Slope of FM", and write down the slope when h=0.5, h=0.2, h=0.1, and h=0.01. What is the slope seem to approach?
Question 6: What happens if we substitute h=0 into the equation?
Question 7: So, we want h to approach 0, but not equal 0, so we can use limits. Write the equation for slope with the correct limit notation, and then solve for the exact slope of the tangent line.