Understanding Calculus Derivatives and Antiderivatives

1. Problem of the Day - Calculator If the derivative of f is given by f '(x) = ex - 3x2, at which of the following values of x does f have a relative maximum value? A) -0.46 B) 0.20 C) 0.91 D) 0.95 E) 3.73

2. Problem of the Day - Calculator If the derivative of f is given by f '(x) = ex - 3x2, at which of the following values of x does f have a relative maximum value? A) -0.46 B) 0.20 C) 0.91 D) 0.95 E) 3.73

3. Recall the Power Rule ∫xn dx = xn + 1 + c where n≠-1 n + 1 What is the antiderivative if n is -1?

4. Recall the Power Rule ∫xn dx = xn + 1 + c where n≠-1 n + 1 What is the antiderivative if n is -1? We define it to be the natural logarithmic function.

5. Logarithms were invented by the Scottish theologian and mathematician John Napier.

6. is area under curve from 1 to x and is positive is area under curve from 1 to x and is negative

7. What if x = 1?

8. What if x = 1? It is 0 because upper and lower limits are equal thus ln(1) = 0.

9. Properties of the Natural Log Function Domain (0, ∞) Range (-∞, ∞) Continuous Increasing 1 to 1 Concave Downward

10. General Log Properties ln(1) = 0 ln(ab) = ln(a) + ln(b) ln(an) = n ln(a) ln(a/b) = ln(a) - ln(b) logb a = x means bx = a

11. General Log Properties ln(1) = 0 ln(ab) = ln(a) + ln(b) ln(an) = n ln(a) ln(a/b) = ln(a) - ln(b) Example

12. Remember - when rewriting logarithms check to see if the domain of the rewritten function is the same as the original Example ln x2 the domain is all reals except x = 0 2ln x is all positive reals

13. Logarithms are defined in terms of a base number Common logs have a base of 10 because log10 10 = 1 (101=10) Natural logs have a base of ≈2.71828182846. . . which has been defined as e (Euler Number) because ln e = =1

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