Math 1304 Calculus I

1. Math 1304 Calculus I 3.9 – Related Rates

2. For your reference: 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)

3. Steps for Solving Related Rate Problems • Read the problem carefully • Draw a diagram if appropriate and possible • Introduce notation. Assign variable names • Express the given information in terms of functions and their derivatives • Write down the formulas that express these relationships • Use the chain rule to help with implicit differentiation • Substitute and simplify where appropriate • Solve the resulting equations as you can • Verify that you answers are in the feasible range of values for the problem.

4. Examples • Types of problems • Volumes of spheres, boxes, and cones • See formulas inside front cover • Examples 1, 3 • Ladders and roads • Use distance formula (Pythagorean formula) • Examples 2, 4 • Rotating things (Search lights, etc.) • Use trigonometry • Example 5