Calculus Answers

(a) Evaluate∫[

𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙.

(b) Use MATLAB to generate some typical integral curves of 𝑓(𝑥) =

𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙over the interval (−5,5).

Find an equation of the tangent plane to the surface at the given point. f(x, y) = x2 − 2xy + y2, (1, 5, 16) with maple lab please

Find an approximate value of the double integral below where 𝑅 is the rectangular region having

vertices (−1, 1) and (2, 3). Take a partition of 𝑅 formed by the lines 𝑥 = 0, 𝑥 = 1, and 𝑦 = 2, and take (𝑢𝑖

, 𝑣𝑖) at the

center of the 𝑖th sub region.

∬(3𝑦 − 2𝑥

2)𝑑𝐴

𝑅

For the following exercises, draw the given optimization problem and solve.

1) Find the volume of the largest right circular cylinder that fits in a sphere of radius 1.

Find the volume in the first octant bounded by x+y+z=9, and the inside cylinder 3y=27-x^3

Show whether the following functions are uniformly continuous on the given domain.

1. F(x)=x^3 on [-1,1]

2. F(x)= 2x/2x-1 on [1, infinity]

3. F(x)= sinx/x on (0,1)

4. F(x)= 1/x on (0,1)

Find the area bounded by the given curves; y= 1/(1+x^2), x=1,x=-1. sketch the curve indicating the area bounded.

y=sin(2x)/x, x=Pi, dx=.25

find the differentiate

If y=sin(2x)/x and x =Pi and ex=.25 find the differentaite

Find the dimensions of the right circular cone of the greatest lateral area that can be inscribed in a sphere of radius α