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UMD - Study Guide - Midterm

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Exam #2

Study Guide

SSON

Sections: 3.2 3.3, 3.4, 3.5, 3.7 Topces:

* Derivatives * s Product Rule - Quotions Rule - > Chain Rule * Related Rates * * Implicit Differentiation

* Higher Derivatines Commen Derivatives :

f (x) = 1 * d (ex) = ex * d (sind) = cos x* d (lulx)) = x 1 x 20 đ (es) in *

en el lulxl) = kx+0 & (tanx) = secs d (loga (x)) = xtra 470 a secx) = seex tang*

(Cs cx) = escx cotx* d ( cotx) = - csc 3* of (sin "k) - VLXI og ( 603 tk) VAN

(tan+x)= entitats d (at) = ax Incas A

SO

14x2

Chain hule

[ C 40x3]")= n [fvx)]*4'64) lete) = f'()e they

(lu [fear]). er ing 4 ene (sin [80x2]) = f'(x) cos Ef(x)] s. (c) [feed])= = f'(4) sinta] " # ( tantf(x)])f'(6) sec? [yud] z (Sc ]) = f'be) sur Efte]tau (86)) e ( ton" [fes])= ve

Higher Order Derivativess: The nth derivative is densted as The Second derivative is denoted as f (x) = d is defined as

We also discuss several other topics like harmophonic

f"(x) = f(a)(x) = d[ and fix] =(feat is defined as fix) = (+47) the derivative of the

(n-1th derivatie,

* the derivative of

the dentar

the first derivatie

feno)(x).

Jl.

SSO

Quotient Rule:

low

d-high

high

to low

IF ( 94

High cour)

[gb]2 squared * Low "a-high minus high of flow over the

de nounsnator square we go!!! Product hele! dy (dex - glx)) = fulg'(x) + f(x1944) I y (x y z) = n'yz+ yy'z + xyz' Implicit Different sathione * The "trick" is to differential normal ☆ every time you differentiate "y" you took an dy (from chain rule)

Ex!

SESO

(y?+2)(2x) = (x+1) (2y on 24

- (y2+272 (y=+2)(2x) – (x2-1)/243 dole) = 2x[y_tz)?

2x (22+2) - (42+2) (2x)

-(x2-1) (24)

v

Relented

Peates:

* Always Draw a

nie

We also discuss several other topics like m nasiri uc davis

he

picture*

find

formula

Points ABB

are moving

the graph as follow. Suppose dxA ezt

(#3) derivativa (#4) Sohne Jes

Unknown

1

YX

YA YA

Ford rate of charge of distene betwen 143 B when AG?!? *BO 42), t=5!

D = √(x4-753) =(YA - Ys)? Find do? le (Dva-Xo) Alyangan) * (21x2 *s) - )+21 Yn ya) de en opt) We Koner that

it Iz = K2+1 YA = 2 2XA

?

2

T

We know that

dya 24 F #(02-1)+1-2)2) Ž (242-1) ((12)-1) )+241–27 (13) –1).

( (117+ (-112 ) (211) (61) + 2 (-1) (2)) - (it if ( (2010)(8 + (-2)(x))

More probleus ul Solutions below!!!

A 50ft ladder is placed against a large building. The base of the ladder is resting on an oil spill, and it slips at the rate of 3 ft. per minut the rate of change of the height of the top of the ladder above the ground at the instant when the base of the ladder is 30 ft. from the base of the building.

A stone dropped in a pond sends out a circular ripple whose radius increases at a constant rate of 4 ft/sec. After 12 seconds, how rapidly is the area in closed by the ripple increasing?

Organizing information:

50 Don't forget about the age old question of university of toledo biology

• Goal: Find A when t = 12. We use the area formula for a circle.

Claudy Soul

Organizing information:

dy = 3

A = ar2 Differentiate both sides with respect to t:

Plug in

= 4. When t = 12 seconds, r = 4*12 = 48.

• Goal: Find when y = 30. We use Pythagorean Theorem again:

22 +30o = 50° I = 40. And differentiating (notice how the hypotenuse is constant): We also discuss several other topics like savanna butler

20x + 2yy' = 0x'

= -24 = Tyy Plugging in, r' = -30.3 + 40 = -2.25.

A = 27 (48) * 4 = 3841 ft2/sec

20

is decreasing—the ladder

Note: I is negative, that means the distance is slipping down the building.

The radius of a cylinder is increasing at a rate of 1 meter per hour, and the height of the clinder is decreasing at a rate of 4 meters per hour. At a certain instant, the base radius is 5 meters and the height is 8 meters. What is the rate of change of the volume of the cylinder at the instant? Don't forget about the age old question of consider a dht with a mesh overlay topology (that is, every peer tracks all peers in the system). what are the advantages and disadvantages of such a design? what are the advantages and disadvantages of a circular dht (with no shortcuts)?

A spherical balloon is being inflated so that its diameter is increasing at a rate of 2 cm/min. How quickly is the volume of the balloon increasing when the diameter is 10 cm?

Organizing information:

= 2 dt

• Goal: Find when d= 10. We use the volume formula for a sphere, but rewrite it with the diameter Don't forget about the age old question of Which of the outer planets does each of the following apply to (the number in parenthesis is how many planets should be included in the answer)?

Organizing information:

drdh

• =1, dt

= -4

• Goal: Find when r = 5, h = 8. We use the volume formula for a cylinder

dV

dd

dt

dt

V = arah Differentiate both sides with respect to t: (you have a product rule on the right side)

av = (nr?) + fen + (251

Plug in

= 1, h = -4, r = 5 and h = 8.

Differentiate both sides with respect to t:

= ((5)?) *(-4) +8* (27(5)(1)) = -2007 + 807 = -120m/hour

dy

3

dt

Plug in d = 2 and d = 10

= 3610)+2 = 1007cm» /min

A person who is 6 feet tall is walking away from a lamp post at the rate of 40 feet per minute. When the person is 10 feet from the lamp post, his shadow is 20 feet long. Find the rate at which the length of the shadow is increasing when he is 30 feet from the lamp post.

dir

The diagram and labeling is similar to a problem done in class. Organizing information:

40, when x = 10, s=20

• Goal: Find s when I = 30. We set up a ratio of similar triangles.

+S

S

The height of the pole is a constant. We solve for h by using that when <= 10, s=20.

10+20 20 ho

6(30) = 20K

h=180/20 = 9 Now rewrite our orginal ratio equation with the constant height solved for:

+S s

96 60 + 6s = 9s

60 = 3s Differentiate both sides with respect to t and solve for ds

Plug in d = 40, and solve for ds

di = 80ft/min