Estimating the Derivative at a Point from Graph

Geometric Meaning of the Derivative of a Function at a Point

Recall that the derivative of a function at a point is a number which is the slope of the tangent line line to that function at that point. In the app: Start by entering a formula for the function in the input box for f(x). Adjust the point by entering its x-value in the input box for a or by adjusting the slider.

Approximation Method 1

In the app: Adjust point B to get the black line to be a tangent line to the curve at point A. Determine the slope of this line and enter it in the input box for the estimate m_1.

Approximation Method 2

Use a symmetric difference quotient to approximate the derivative. Pick two points on the function equally spaced horizontally on both sides of point A. Use the coordinates of these two point and the slope formula to compute a symmetric difference quotient. Enter this value in the input box for the estimate m_2.

Checking Your Answer

Are your two estimates close to each other? Which one is closer to the actual derivative? What is the relationship between the sign of the derivative and the shape of the graph? Did you get the sign correct?