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Unformatted text preview: Project 14
1. Rewrite the function f (x) = ln| x | as a piecewise deﬁned function. Draw its graph on the
axes below. f (x) = Calculate the slopes of the tangent lines to f (x) at x = 2 and x = −2. Draw these two
tangent lines on the axes above. Find the derivative of g (x) = ln(ax) where a is any positive number using chain rule. Does
the answer surprise you? To make sense of it, note that ln(ax) = ln(a) + ln(x). What then
is the relationship between the graphs of ln(x) and ln(ax) ? Why then should they have the
same derivative (think in terms of slope)? 2. Examine the function f (x) = sin(ln(x)). Show that x2 f (x) + xf (x) + f (x) = 0 for every
x > 0. (We would say that f (x) solves this differential equation on the interval x > 0, since
the right-hand side and left-hand side are the same function on that interval.) Although this function is a solution to a very basic differential equation, it is a fairly “wild”
function. Find the x-values of its horizontal tangent lines and show there are inﬁnitely many
of them on the interval ( 0, 1 ) . (Try graphing on a calculator and note what happens as you
zoom on smaller and smaller intervals [ 0, x ]) 3. Find where the functions have horizontal tangents:
4x − 12 ln(8x − x2 ) x2 (ln(x))2 4. For which ONE of these functions would logarithmic differentiation truly be beneﬁcial?
xe x2 2 ln(x2 ex ) x2 + ex 2 x2
ex2 Calculate the derivative of that function. 1 2 5. Examine the function f (x) = xe 2 (x−2) (x + 2)3 .
Use logarithmic differentiation to calculate the derivative of f (x) . At what values does logarithmic differentiation fail? What is the slope of the tangent line to f (x) at x = 2? x = 0? Where does f (x) have horizontal tangent lines (add up your fractions. . . )? ...
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