As students start their study of Calculus AB and BC they learn to study small changes made along a function in both the x and the y are labeled Δx and Δy and that changed along a tangent line called differentials and use the symbols dx and dy.
This activity helps students see that if they are looking to approximate a change made along a function that they approximate this change by find the change along a tangent line. Since the tangent line is linear and the function probably is not linear, this approximation is given the term a linear approximation.
Two activities are included in the lesson.
• The first activity explores the function y = 2 square roots of x in a region near the point (1, 2). Using a graph the students find the slope of the function at (1, 2) and draw the appropriate tangent line of the graph at (1, 2). Using the graphs of the function and its tangent line students find three approximate the changes made along the function and compare it to three approximations made along tangent line. They observe that as Δx becomes smaller, the absolute value of the difference between Δy and dy approaches zero.
• The second activity explores the function y = 12x^2. After drawing a graph of both the function and its tangent line at (0.5, 3), students enter both equations in their graphing calculator and then use the zoom-in feature to zoom in on the point (0.5,3) to notice that the two lines almost become the same line in a small window around (0,5, 3). The students also build a set of table values for both the function and its tangent line when x is very close to 0.5. Students are asked to describe how close the x values need to be to 0.5 to keep the difference between the |dy-Δy| less than 0.2 or 0.1.
• Students are then asked to describe how their understanding of differentials and linear approximations have been enhanced by completing the two activities.
A full set of answers has been included.
You might be interested in the Activity Sheet on Differentials that is also available in my store.
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