16 3
V
r
3
dV
2 dr
 16r
dt
dt
1
y  3x 
x
dy
dx 1 dx
 6x
 2
dt
dt x dt
y
x

100 70
dy 10 dx

dt
7 dt
2
At a glance
4.1 Related Rates Problem p. 18
1. Identify all GIVEN Quantities
2. Write an equation
3. differentiate with respect to time.
4. AFTER differentiating , substitute in all
values and solve for what is asked.
1.
2.
3.
No context, just an equation.
Geometry
Pythagorean
A function on a closed
interval will have only 1
absolute (global) max. and
1absolute min.
 A function can have many
relative max/ min.

If your function is continuous on a closed interval, then there are
4 easy steps to find the absolute (global) extrema.
Steps:
1: Find critical numbers. (derivative is 0 or undefined)
2: Make a T-chart of critical values and endpoints
(plug into original function)
3: Identify the absolute max.
4: Identify absolute minimum.
We use a number line:
Local (relative) max: when f’(x) changes from + to –
at some x value.
Local (relative) min: when f’(x) changes from - to +
at some x value.
Mean Value theorem: If f is contin. on the closed interval, and
differentiable on the open interval then: there exists a number c
between a and b such that:
f b   f  a 
f '(c) 
ba
“Derivative equals
the slope of the secant
at some point”
Mean Value theorem: If f is contin. on the closed interval, and
differentiable on the open interval then: there exists a number c
between a and b such that:
f b   f  a 
f '(c) 
ba
IF: (you must state)
1. f(x) is continuous on the closed interval.
2. f(x) is differentiable on the open interval.
Then:
f b   f  a 
3. There exists a c such that f '(c) 
ba
4. (not always) solve for c
L’Hospitals Rule: For limits of
indeterminate forms 0 or  
0

 Note this is NOT a quotient rule!

Ch 4 at a glance-ch 3 at a glance,
do as much as you need

16 more days!