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Show that the MacLaurin's series for f (z) =sin z is given by
z z 3 s "' (- I)" z211+J
f(z) =z--+- ....+L ( ) +...
Evaluate each of the following limits. Simplify all answers completely. Show all work toward your answer or provide an explanation for how you arrived at your answer.
1) lim x--> (x^2 -1)/(x^2 -8x+7)
2) lim x-->0^negative(-) (2)/(tan(x))
3) lim x--> infinity (cos(t))/(e^3t)
4) lim x--> (-)infinity (-3x^3 +5x^2 -4)
Find each of the following derivatives. Simplify all answers completely. Show all work toward your answer or provide an explanation for how you arrived at your answer.
1) f(x)=sqroot e^2x + 8x^2 e^x
2) g(x)= xsin(x)/1+cos(x)
3) h(x)=e^-x sin(x)
4) y=sec(x)tan(x)
5) f(t)= ((t-1)(2t^2 -1))/(t^3 -1)
A cone is generated when the region is bounded by the line x=y and the vertical line x=0 and x=r is rotated about the x-axis . Use the pappus theorem to show that , Surface area of the cone is given by S=β2ΟrΒ²
Find the area A(R) of the region R described below.
1. R is the region bounded by y = 2x + 3, the x-axis and the line x=5 .
2. R is the region bounded by y=x^{2}+1 , the lines y=2 and y=5
3. R is the region bounded by x=y(8-y) and the y-axis.
4. R is the region bounded by y = x and y=x^{3} .
5. R is the region bounded by y=2-x^{2} and y=x^{2}-6
Show that β2π β«2π π₯ 2 sin8 (π π₯ )ππ₯| β€ 16π3 /3
Identify the domain of the following functions.
a) π(π₯) = sinβ1 (3π₯ β 1)
b) π(π₯) = [log(sinβ1 (βπ₯ 2 + 3π₯ + 2))]
c) π(π₯) = 1 β(π₯β2)2
d) π(π₯) = 1 π₯ββ(π₯+2)
e) π(π₯) = ln|π₯ + 3| β 5
A poster must have 32 square inches of printed matter with margins of 4 inches at the top, and 2 inches at each side. Find the dimensions of the whole poster if its area is MAXIMUM.
When two given functions, π(π₯) = π₯ 2 + 3 πππ π(π₯) = 2π₯ β 5, are divided with each other, a new third function is obtained. Find (π/π)(π₯) and identify its domain
#339914 on Dec 2023