Calculus Answers

Let f(x) = x^2 + 6x. Use the definition (ε − δ method) to show that
lim
x→−3 f(x) = −9

The velocity vector in a fluid flow is given by V = 2x^3 i - 5x^2 y j + 4t k. Find the velocity and acceleration of a fluid particle at (1, 2, 3) at time , t = 1.

Evaluate ∫∫D xydA, where D is the region bounded by x = y^3 and y = x^2.

Consider the curve defined by 2y^2-x^2y=3.

a. Show that dy/dx=2xy/3y-x^2

b. Write an equation for the line tangent to the curve at the point (1, -1).

c. Show that there is a point P with x=coordinate 0 at which the line tangent to the curve P is horizontal. Then find the y coordinate of P.

https://us-static.z-dn.net/files/db5/e56bcd84953d9511cb78ca02bcbd9bbc.jpeg

Find the most general antiderivative for the function 6x−−√3−5x2−−√3.
Antiderivative =

Let f(x)=8x−2ex.
Enter an antiderivative of f(x)

Identify the curves (lines, parabola, etc.) generated by illustrating the graphs.

5. Consider the region enclosed by the curves y=√x, y=6-x, and the x-axis. Rotate this region about the x-axis and find the resulting volume.
6. Consider the region enclosed by the curves y=√x, y=6-x, and the x-axis. Rotate this region about the y-axis and find the resulting volume.
7. Consider the region enclosed by the curves y = x^2 − 2x and y = 3. Rotate this region about the line y = 3 and find the resulting volume.
8. Consider the region enclosed by the curves x = y^2 and x = 2 − y^2. Rotate this region about the line x = 3 and find the resulting volume.

Polar Coordinates: Solve the problems by testing the symmetry, plotting and tracing the curves, and
computing the area, with limits from 0 to 2π, of the following equations

1. r = a cos 2θ
2. r = a sin 3θ
3. r = a (1 + sin θ)
4. r^2 = a^2(cos 2θ)

Determine whether the following functions
i). f(x) = 2x3 - 3x + 1; [-2; 2]
ii). f(x) = e^x, [0; log 4]
iii). f(x) = log 2x, [1; e]
iv). f(x) = sin^(-1) x, [0; 1/2]
meet the conditions of the Mean Value Theorem on the interval. If so, nd the
point(s) guaranteed to exist by the theorem