1. For f(x) = (1-x)/(1+x) and g(x) = x/(1-x), find the simplified form for f[g(x)] and state the domain?
2. Evaluate lim x -->1 ((√x^2 +8) -3))/(x-1)
3. A particle moves on a line away from its initial position so that after t hours it is s = 4t^2+t miles from its initial position. Find the average velocity of the particle over the interval [1,4].
4. Use the table to evaluate d/dx [g[f(3x)]] at x = 1
x 1 2 3 4
f(x) 6 1 2 2
f'(x) 6 1 10 2
g(x) 1 4 4 3
g'(x) 4 5 7 -4
5. Find the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(2x) in the interval [0,2π]
6. Find dy/dx for 4 - xy = y^3
7. A 160 inch strip of metal 20 inches wide is to be made into a small open trough by opening two sides on the long side, at the right angles to the base. The sides will be the same height, x. If the trough is to have a maximum volume, how many inches should be turned up on each side?
8. The radius of a 10 inch right circular cylinder is measured to be 4 inches, but with a possible error of +- 0.1 inch. What is the resulting possible error in the volume of the cylinder?
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#1
f(x) = (1-x)/(1+x) and g(x) = x/(1-x)
f(g) = (1-g)/(1+g)
= (1-(x/(1-x))/(1+(x/(1-x))
= 1-2x
while the domain of 1-2x is all reals, g is undefined for x=1, so that must be excluded from the composite as well.
#2 multiply by the "conjugate"
(√(x^2+8)-3)/(x-1) * (√(x^2+8)+3)/(√(x^2+8)+3)
= (x^2+8-9)/((x-1)(√(x^2+8)+3))
= (x^2-1)/((x-1)(√(x^2+8)+3))
= (x+1)/(√(x^2+8)+3))
Now take the limit, and you get 2/6 = 1/3
#3 the average velocity is total distance/time
= (s(4)-s(1))/(4-1)
#4
d/dx g(f) = g'(f(3x)) * f'(3x) * d/dx (3x)
= g'(2) * f'(3) * 3
= 5*10*3
= 150
#5
f(x) = 2x + sin(2x)
f' = 2+2cos(2x)
f'=0 when cosx = -1
...
#6
4 - xy = y^3
-y - xy' = 3y^2y'
y'(3y^2+x) = -y
y' = -y/(3y^2+x)
#7
v = 160x(20-2x) = 320(10x-x^2)
dv/dx = 320(10-2x)
max v when dv/dx=0, at x=5
#8
v = πr^2h
dv = 2πrh dr
Now just plug in your values for r and dr to get dv