Lecture Notes: Wednesday, October 28, 1998

Eighth Homework Assignment - for discussion at recitations on Tuesday November 3, and Thursday November 5.

Chapter 5

• Section 4 (Differentiation of Exponential Functions)
• Section 5 (Differentiation of Logarithmic Functions)

FACT: as h -> 0, (e^h - 1)/h -> 1.

From this we derived the fact that the derivative of the exponential function (base e) is the exponential function itself.

(d/dx)(exp(x) = exp(x).

Differentiate the functions: e^(x^2), e^(2*x), 2^x, e^ln(x)

Now we can do max/min, graphing, tangent (etc.) problems with exponential and logarithmic functions. Be sure to work the assigned problems from Old Final Exams.

Inverse Functions again. f(g(x)) = x, g(f(x)) = x. Differentiating both sides of the latter gives: g'(f(x)) = 1/f'(x). We use this technique to get a formula for the derivative of ln(x).

Differentiating x = exp(ln(x)) gives: 1 = exp(ln(x))*ln'(x), from which we determined

(d/dx)(ln(x) = 1/x.

Examples: Differentiate: ln(x^2+2), log[10](x) (In doing the second one we derived the fact that

log[10](e) = 1/ln(10).

Logarithmic differentiation. Use it on

• (x^2+1)(x-4)(x^3+3x)(x)
• x^(x^2)

In MAPLE, we looked at the graphs of e^x, e^(-x), and e^(-x^2).

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EXAM #3 Wednesday, October 28. We are in DRL A1 and A8. Please be in your seats by 6:20.

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Maple Demonstration #8 Please work through this demonstration on your own. Make particular note of the fact that there is a unique exponential curve passing through a given pair of points. Also note use of the differential equation solver.

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