Math 1304 Calculus I

1. Math 1304 Calculus I 4.09 – Antiderivatives

2. Antiderivatives • A function F(x) whose derivative is a given function f(x) is called an antiderivative of f(x). • An antiderivative of zero is any constant. • Two anitderivatives of a given function differ by a constant.

3. Notation

4. Example • Power - please verify

5. Recall: A good working set of rules • Constants: If F(x) = c, then f’(x) = 0 • Powers: If F(x) = f(x)n, then F’(x) = n f(x)n-1 f’(x) • Exponentials: If F(x) = af(x), then F’(x) = (ln a) af(x) f’(x) • All trigonometric functions: If F(x) = sin(f(x)), then F’(x) = cos(f(x)) f’(x) If F(x) = cos(f(x)), then F’(x) = - sin(f(x)) f’(x) • Hyperbolic functions • All inverse trig functions • Scalar mult: If F(x) = c f(x), then F’(x) = c f’(x) • Sum: If F(x) = g(x) + h(x), then F’(x) = g’(x) + h’(x) • Difference: If F(x) = g(x) - h(x), then F’(x) = g’(x) - h’(x) • Multiple sums: derivative of sum is sum of derivatives • Linear combinations: derivative of linear combo is linear combo of derivatives • Product: If F(x) = g(x) h(x), then F’(x) = g’(x) h(x) + g(x)h’(x) • Multiple products: If F(x) = g(x) h(x) k(x), then F’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x) • Quotient: If F(x) = g(x)/h(x), then F’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2 • Composition: If F = fog is a composite, defined by F(x) = f(g(x)) then F'(x) = f'(g(x))g'(x)

6. Rules for Antiderivatives • Each of these transforms into a rule for antiderivatives.