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
[ 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
* the derivative of
the first derivatie
IF ( 94
[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)
(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)
* Always Draw a
We also discuss several other topics like m nasiri uc davis
the graph as follow. Suppose dxA ezt
(#3) derivativa (#4) Sohne Jes
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
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?
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.
dy = 3
A = ar2 Differentiate both sides with respect to t:
= 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
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?
= 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)?
• =1, dt
• Goal: Find when r = 5, h = 8. We use the volume formula for a cylinder
V = arah Differentiate both sides with respect to t: (you have a product rule on the right side)
av = (nr?) + fen + (251
= 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
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.
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.
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:
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