Mathematics 133 -- Intensive Calculus for Science 1
Discussion 5 -- Practice With Derivatives
October 31, 2001
Background
We have now seen the three general ``shortcut rules'' for differentiating
- Products: d/dx(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)
- Quotients: d/dx(f(x)/g(x)) = (g(x)f'(x) - f(x)g'(x))/(g(x))2
- Compositions: d/dx(f(g(x))) = f'(g(x))g'(x)
Today we will practice using these and use some material from
Chapter 2 on the meaning of the signs of f' and f''.
Discussion Questions
All of the following questions refer to
f(x) = x2 e-5x
Do not use a calculator (until the very end, to check your graph)
- A) What rule(s) could you use to compute the derivative
of f(x). Find this derivative by any one applicable method
and simplify by factoring as
much as possible.
- B) Over which intervals is f(x) increasing? decreasing?
- C) Where are the turning points of f(x) and what type is
each one of them -- ``peak'' (maximum), ``pit'' (minimum), or neither?
Explain how you can tell.
- D) Now find the second derivative of f(x). Again,
simplify by factoring as much as possible.
- E) At which x is f''(x) = 0?
Over which intervals is the graph y = f(x) concave up?
concave down?
- F) Using all the information above, sketch the graph
y = f(x).
Assignment
Group write-ups due in class on Tuesday, November 6.