Section 4.6 rates and related rates
Part 1. Rate is derivative
vVhen you see ~, you should know it means: 6 is the variable , 0 is a function of 6 , and
~ is the derivative.
# 1. (page 226, #36, see Figure 4.87) A lighthouse is 2 km from the long, straigh
Section 2.<J interpretation of derivatives
1. f' (a) tell s us how fast .f (:c) is changing a t x =a , but how fast? See the fon11t!la
J(a + h)- f(a)
it means from .c = a to x = a+ h, f( .c) changes by app1'0xintately f' (a)h. so at .1' = a,
there are two ways to solve it .
way 1 (you see e, so you can use ln to make life easier because ln and e will cancel. Similar
to way 1 in the previous example, we apply ln to both sides directly)
use formula. ln(ab)
= ln(a) + ln(b),
then ln and e
< e ex
-~ . . -"'- - -
_ 'l1"l_; _
e: - · e b_· · .oc at: g p · g o d i o ) e · . A·
p . . s: .o· ices
that the mold has grown to cover an area of 5 cm 2 . At 7pm . she return and find he mold
now covers an area of 7 em 2 .
Sect ion 3.4 , 3.5
Part 1. Derivative rules for sinusoidal functions and log function
(1) (sin (x)' = cos(x), (cos(x)' = sin(x), (tan(x)' = cos~ (x)
(Note that we write (cos(x) 2 as cos 2 (x), similarly for sin 2 (x).)
(2) (ln (x) = ~ , (log(x)' = x ln~
Section 5.2 definite integral
Part 1. How do we calculate definite integral?
( 1) When a :; b,
f (x) dx = area above x-axis - area below x-axis.
(2) \Nhen a> b, use
f(x)dx = - Iba f(x)dx to swap the limits
IJ(x)ldx =total area.
(3) When a:; b,
,._ 2 )
o .:.: s a nd 3 rob lems in total )
:.·:o oi :~;: ~: T e .-\·kward Turtle is ridi ng a mini ferris wheel! The wheel has radius 1
:_ e e b is lifted off the gro und , so that even when he is at the lowest point of t he ride.
he .-\ \-k\·ar
Let q be the number of items produced. Say the cost function of producing q items is
C(q), the revenue function is R(q) , then the profit function n(q) = R(q)- C(q) .
C(O) is called the fixed cost, which is the cost needed (for example, mach