F.5 maths 問題,請數學高手快進~!!THX

1. write the partial fraction decomposition of the rational expression
6x/(x+4)(x-2)
6x/(x+4)(x-2) = A/(x +4) + B/(x-2)
6x/(x+4)(x-2) = [A(x-2) + B(x+4)]/(x +4)(x-2)
6x/(x+4)(x-2) = [Ax+ Bx-2A+ 4B]/(x +4)(x-2)
6x/(x+4)(x-2) = [(A+ B)x+ (4B -2A)]/(x +4)(x-2)
A+B = 6
4B – 2A= 0
A = 2B
2B+B = 6
B = 2, A = 4
6x/(x+4)(x-2) = 4/(x +4) + 2/(x-2)

2. evaluate the summation
5
Σ (i+5)^-1
i=2
=1/(2+5) +1/(3+5) +1/(4+5) + 1/(5+5)
=1/7 + 1/8 + 1/9 + 1/10
=1207/2520

3.of the given values of x, identify those at which the function is continuous
f(x)= x+7/(x-2)(x-8); x=-7,0,2,8
a. -7,0,2,8
b. -7, 2, 8
c. 2,8
d, -7,0
vertical Asymptote at x = 2, x = 8
answer (d) -7 and 0

4. solve the problem
the surface area of a right circular cylinder of height 4 feet and radius r feet is SA = S(r)= 2 πrh+2πr^2. find the instantaneous rate of change of the surface area with respect to the radius r at r =4
dS/dr = 2 πh+4πr = 2 π(4)+4π(4) = 8 π +16 π = 24 π

answer (c)
a.20π
b.16π
c.24π
d.64π

5. approximate the area under the curve and above the x-axis using n rectangles.let the height of each rectangle be given value of the function at the right side of the rectangle.
f(x)=3x^2 - 2 form x=1 to x=5; n=4
When x = 1, f(1) = 3(1)^2 – 2 = 1
When x = 2, f(2) = 3(2)^2 – 2 = 11,(1+11)/2 = 6
When x = 3, f(3) = 3(3)^2 – 2 = 25, (11+25)/2 = 18
When x = 4, f(4) = 3(4)^2 – 2 = 46, (25+46)/2 = 35.5
When x = 5, f(5) = 3(5)^2 – 2 = 73, (46+73)/2 = 59.6
4 rectangles,
from x = 1 to 2, area A1 = 6 x 1 = 6
from x = 2 to 3, area A2 = 18x 1 = 18
from x = 3 to 4, area A3 = 35.5= 35.5
from x = 4 to 5, area A4 = 59.5 x 1 = 59.5
area = A1 + A2 + A3 + A4 = 6 + 18 +35.5 +59.5 = 119