Calculus Answers

If In=int_0 to ♾️ {(e^-x)(sin^n) (x) dx } , prove that (1+n^2)In=n(n-1)In-2 for n≥2 .

Find the equations of the tangents to the graph of y=x+1/x that are parallel to y+2x=0

Find the equation of the tangent to at the point where , in f(x)=x^2+4x-5

standard form.

Find the equation of the tangent line to a curve y=-x²-1 that is parallel to the line 2x+y=6.

Find the slope and the equation of the tangent of the curve at the given point. Show your solution.

1) y = 2x² -x + 2 (-1, 5)

2) y = x² + x + 1 at (-2, 3)

Find the slope of the curve at the given point. Show your solution.

1) y = 4 - 3x² at (-1, 1)

2) y = x² – 5 at (1, -4)

3) y = 2x² – 5 at (2, 3)

A cylindrical container has one end opened and the other end closed. It has a circular base of radius r cm. Given that the total surface area of the container is 200π cm2.

(a) Show that the volume of the container is V = 100πr - πr^3/2 cm3

(b) Find the maximum value of V.

b) Find whether the following series are convergent or divergent √ 𝟏 /𝟒 + √ 𝟐 /𝟔 + √ 𝟑 /8 + ...

Find the dimensions of a right circular cone of minimum volume V that can be

circumscribed about a sphere of radius 8 inches.