Unit 3

Question 1

concave up, increasing, positive

Answer 1

if second derivative is positive, f is concave up
if derivative is positive, f is increasing

Question 2

formal definition of a derivative

Answer 2

f’(x) = lim (h->0) f(x+h)-f(x) / h

Question 3

equation of a line tangent to a curve

Answer 3

y - y1 = m(x-x1)

Question 4

points of inflection occur when

Answer 4

f(x) changes concavity, so when f’‘(x) changes signs (ex. +-)
- when f’(x) changes direction (ex. increasing to decreasing)

Question 5

particle at rest when

Answer 5

v(t) = 0

Question 6

displacement

Answer 6

final point - initial point

Question 7

critical numbers

Answer 7

any x-value(s) where a relative max or min might exist

Question 8

Rolle’s theorem

Answer 8

if the slope of the endpoints is 0, then somewhere on the interval [a, b] the derivative must be 0
- must prove f(x) is continuous and differentiable on the interval
- f(a) must equal f(b)
- set f’(c) = c to find c, and check that it is on the interval, which it should be

Question 9

OPTIMIZATION PROBLEMS

Answer 9

1. create a system of equations
2. solve one equation for a variable and substitute into the other equation
3. take the derivative and set it = 0

Question 10

distance formula

Answer 10

[ (x-x1)^2 + (y-y1)^2 ]^1/2

Question 11

tangent line approximation (c, f(c))

Answer 11

y = f(c) + f’(c)(x-c)

y = b + mx