Calculus Answers

Use the Gamma function to evaluate int_{0}{1)x^{4}{1-x}^{3}dx

Let ω0<π/2. The value of ∑[sin(ω0n)/πn]^4 equals

For the function f(x)=x^3 - 6x^2 - 63x +124, Determine the following:
i. The turning points
ii. Sketch the graph of the function.
iii. The x-values where the function is increasing and decreasing

9.Evaluate the \\(\\frac{d ^{3}f}{d x^{3}}\\) of \\(f(x)= sin (x) cos (x)\\)
a.\\(\\frac{d ^{3}f}{d x^{3}}=-2\\left(Cos^{2} (x)+sin^{2} (x)\\right)\\)
b.\\(f\'(x)=5x^{4}-\\frac{1}{2}x^{\\frac{1}{2}}+ \\frac{1}{2x^{2}}, 20x^{3}-\\frac{3}{4}x^{-\\frac{1}{2}}- \\frac{1}{x^{1}}, 100x^{2}-\\frac{3}{8}x^{-\\frac{3}{2}}+ \\frac{3}{x^{4}}\\)
c.\\(\\frac{d ^{3}f}{d x^{3}}=-4\\left(cos^{2} (x)-sin^{2} (x)\\right)\\)
d.\\(\\frac{d ^{3}f}{d x^{3}}=-4\\left(tan^{2} (x)-cos^{2} (x)\\right)\\)

10.Determine whether the Rolle\'s theorem can be applied to \\(f\\) on the closed interval [a, b] . If can be applied, Find the values of \\(c\\) in open interval (a, b) such that \\(f\'( c) = 0\\), \\(f(x)=\\frac{x^{2}-2x-3}{x+2}, [-1, 3]
a.\\(c=-1\\pm\\sqrt(5)\\)
b.\\(c=-2\\pm\\sqrt(5)\\)
c.\\(c=-2\\pm\\sqrt(5)\\)
d.\\(c=-2\\pm 2\\sqrt(5)\\)

7.Find the number \\(c\\) guaranteed by the mean value theorem for derivatives for \\(f(x)=(x+1)^{3}, [-1, 1] \\)
a.\\(c=1\\pm \\sqrt(5)}\\)
b.\\(c=\\frac{-\\sqrt (3) \\pm 2}{\\sqrt(3)}\\)
c.\\(c=\\frac{-\\sqrt (5) \\pm 2}{\\sqrt(5)}\\)
d.\\(c=\\frac{-\\sqrt (2) \\pm 1}{\\sqrt(3)}\\)

8.Determine whether the mean value theorem can be applied to \\(f\\) on the closed interval [a, b] . If can be applied, Find the value of \\(c\\) in open interval (a, b) such that \\(f(x)=x(x^{2}-x-2), [-1, 1]\\)
a.\\(c=\\frac{-1}{3}\\)
b.\\(c=\\frac{-2}{3}\\)
c.\\(c=\\frac{-2}{5}\\)
d.\\(c=\\frac{-1}{2}\\)

5.For \\(g(x)=\\frac{x-4}{x-3}, we can use the mean value theorem on [4, 6], Hence determine \\(c\\)
a.\\(\\sqrt (112) \\)
b.\\(c=2\\pm \\sqrt(3)}\\)
c.\\(c=-2\\pm \\sqrt(5)}\\)
d.\\(c=3\\pm \\sqrt(3)}\\)

6.Given\\(f(x)=\\sqrt(9-x^{2})\\)
a.\\(\\frac{-3}{8}\\)
b.\\(\\frac{5}{8}\\)
c.\\(\\frac{7}{8}\\)
d.\\(\\frac{-9}{8}\\)

3.Find the two x-intercept of \\(f(x)=x^{2}-3x+2\\)
a.x=-2, 2
b.x= 1, 2
c.x=1, 1
d.x=1, 3

4.Compute the first thrre derivatives of \\(f(x)=2x^{5}+x^{\\frac{3}{2}}-\\frac{1}{2x}\\)
a.\\(f\'(x)=5x^{4}-\\frac{1}{2}x^{\\frac{1}{2}}+ \\frac{1}{2x^{2}}, 20x^{3}-\\frac{3}{4}x^{-\\frac{1}{2}}- \\frac{1}{x^{1}}, 100x^{2}-\\frac{3}{8}x^{-\\frac{3}{2}}+ \\frac{3}{x^{4}}\\)
b.\\(f\'(x)=10x^{4}-\\frac{3}{2}x^{\\frac{2}{2}}-\\frac{1}{2x^{2}}, 40x^{3}\\frac{3}{4}x^{-\\frac{1}{2}}- \\frac{1}{x^{3}}, 120x^{2}-\\frac{3}{8}x^{-\\frac{1}{2}}+ \\frac{3}{x^{4}}\\)
c.\\(f\'(x)=10x^{3}-\\frac{2}{2}x^{\\frac{1}{2}}+ \\frac{1}{2x^{2}}, 20x^{3}-\\frac{3}{4}x^{-\\frac{1}{2}}- \\frac{1}{x^{3}}, 10x^{2}-\\frac{1}{8}x^{-\\frac{3}{2}}+ \\frac{3}{x^{4}}\\)
d.\\(f\'(x)=10x^{4}-\\frac{3}{2}x^{\\frac{1}{2}}+ \\frac{1}{2x^{2}}, 40x^{3}-\\frac{3}{4}x^{-\\frac{1}{2}}- \\frac{1}{x^{3}}, 120x^{2}-\\frac{3}{8}x^{-\\frac{3}{2}}+ \\frac{3}{x^{4}}\\)

1.Given \\f(x)=3x(x-1)^{5}. Compute \\(f\'\'\'(x)\\)
a.\\(f\'\'\'(x)=80(2x-1)^{2}(x-1)\\)
b.\\(f\'\'\'(x)=100(x-1)^{2}(4x-1)\\)
c.\\(f\'\'\'(x)=180(x-1)^{2}(2x-1)\\)
d.\\(2i-j\\)

2.Let \\(f(x)=x^{4}-2x^{2}\\). Find the all \\(c\\) (where \\(c\\) is the interception on the x-axis ) in the interval (-2, 2) such that \\(f\'(x)=0\\). ( Hint use Rolle\'s theorem )
a.(-1, 0, 2)
b.(-1, 0, 1)
c.(-1, 1, 1)
d.(-1, 2, 1)

1.Determine if f is one to one for \\(f(x)=x^{2} – 5x+1)\\ when x = 0
a.almost
b.none of the option
c.yes
d.No

2.Let \\(f(x)=x^{2}-4x +7)\\, find \\(\\frac{f(x+\\Delta x)-f(x)}{\\Delta x})\\
a.\\(2x-\\Delta x+4)\\
b.\\(2x+\\Delta x+4)\\
c.\\(2x-\\Delta x-4)\\
d.\\(2x+\\Delta x-4)\\

Evaluate the\\(\\frac{d^{3}f3{dx^{3}}\\ of \\(f(x)=sin(x)\\)