This problem investigates the concept of a derivative from the geometric perspective of limiting secant lines going to a tangent line. It also allows you to explore the rules of differentiation using Maple.
a. Consider the function
We want to investigate the derivative of f(x) at x = 2 by observing secant lines that pass through the points (2, f(2)) and (2 + h, f(2 + h)) for different values of h. The slope of the secant line is given by
and this line always passes through the point (2, f(2)). Find the equations of the 4 secant lines (in slope-intercept form) using h = 0.5, 0.2, 0.1, and 0.05. Use Excel to graph f(x) and the 4 secant lines for x in the interval [1, 2.95]. Limit the range, so that y is in the interval [-10, 20].
b. Find the general formula for the slope of
the secant line through x
= 2 and x
= 2 + h. The slope of the
tangent line is found by taking the slope of the secant line and letting h
tend toward zero. This becomes the derivative of
f(x) at x
= 2. Find the derivative of f(x),
f '(x),
at x = 2.
Find the equation of the tangent line at x
= 2, then use Maple graph
f(x) and the tangent line for
x in the interval
[0,3]. (Use the same limits on the range as in Part a.).
c. Differentiate
the following functions using Maple. You are just about to learn a collection
of differentiation rules, so you should be looking for patterns to help you
in the future as we learn more about derivatives.
Plot a second graph in Maple for both,
f(x) and the tangent line, for a smaller
interval about the point of tangency, choose [2-0.2,2+0.2]. Repeat this zooming process for
an even tighter interval [2-0.02,2+0.02]. What can you conclude from this process?
Explain. (Note that you must include 3 graphs for this part and you must choose your
y ranges appropriately [i.e. tight around the graph of the functions]).
i.
f(x) = x2 + 3x -5
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ii.
f(x) = (x2 + 3x -5)4
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iii.
f(x) = e-3x
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iv. f(x)
=
ln(x+4)
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