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Derivatives and Rate of Change of Curves

See the attached file.
1: Both forms of the definitions of the derivative of a function f at number a.

2: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2ft/sec how fast will the foot of the ladder be moving away from the wall when the top is 5ft above the ground?

3: y': X^2 - 2XY + Y^3 = C

4: Coffee is poured into a cup at the rate of 2cm^3/sec. The bottom of the cup has a radius of 2cm the top of the cup has a radius of 4cm and the cup is 6cm tall. When the coffee is halfway to the top (3cm deep), how fast is the coffee level rising?(hint imagine that the cup is a cone going all the way down to a point instead of being truncated.

5: A particle is moving along a line with its position in meters given by the function
s(t)=1/((9-t)^2)
where t is in seconds. what is the acceleration of the particle after 5 seconds?

6: Find the equation of all the lines which are tangent to the ellipse 3x^2+4y^2=1 and have slop 28/3

7: Show that the two families of curves x^2+(y-c)^2=c^2 and
(x-k)^2+y^2=k^2 are orthogonal trajectories of each other.

8: f(x)=1+x-2x^2
Use either of the two of the definitions of the derivative to show that f'(1)=-3(may need you to do both definitions to help me out pls)

9: compute the derivatives of the functions below:
A: g(x)=(x-1)^2 sin(x)
B: h(x)=(x-3)/(x+1)
C: f(x)=(tan(x))/(1+ cos(x)
D: g(x)=(x+(1/x))^(2/3)
E: f(x)=cos(sin^2 x).

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Please see the attached file for the complete solution.

1: both forms of the definitions of the derivative of a function f at number a.
Solution:
Let f is a real valued function defined on an interval I containing the point a. The derivative of function f at a is denoted by f'(a) and is defined by
, provided the limit exists and finite.
Alternately, we can define the derivative of f at the point a as
, provided the limit exists and finite.

2: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2ft/sec how fast will the foot of the ladder be moving away from the wall when the top is 5ft above the ground?
Solution:

Let the top of the ladder is at yft from the bottom and the bottom of the ladder is at a distance xft from origin. Given when y=5ft. We have the length of the ladder always constant, hence we can write . Differentiating both sides w.r.t 't' we get
. At y=5ft, we have x=12ft(since 52+122=132)
Hence . So the foot of the ladder moves at a rate of ...

Solution Summary

Derivatives and Rate of Change are investigated. The solution is detailed and well presented.

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