# calculus

1. ### Finding the area enclosed by the locus of the vertex of the rectangle at which the normals meet.

Let a and b be the lengths of the semimajor and semiminor axes of an ellipse respectively. Draw a rectangle whose two sides are tangent to the ellipse and the other two are normal to the ellipse. Now how to find the area enclosed by the locus of the vertex of the rectangle at which the...
2. ### Partial Derivative Equation as solutions

I have to solve PDE as ODE, but my lecturer didn't give me enough help and tutorial, to solve these problems. I searched YouTube and other websites to find a clue how to solve them but I find no clue. Please help! The problems are: i have to solve PDE as ODE where u = u(x,y) u_y + u = e^{xy}...
3. ### Optimal distribution?

Suppose we have a young female population of 1 and the probability young females survive and reach older age (lets say >50) is 'k'. While young, females have a fertile potential of b1. While old, females have a fertile potential of b2. Thus lifetime fertile potential for females is 1*b1 +...

Does anyone know how to find the limit of this problem? Your help would be much appreciated!
5. ### Calculus word problem [am I solving this problem right so far?]

Please check the image attached to see the problem and the work I completed. Thank you in advance!
6. ### U-subsitution

Can someone please explain the u-substitution? I don't see a tan(x/2) anywhere in the integral expression.

8. ### Derivative of ln(-x)

Can someone please explain to me why the derivative of ln(-x) is 1/x? Thanks!
9. ### An integral inequality

I spent a day trying to prove this inequality, but I didn't finish it. Hopefully someone will prove it. f(x) and g(x) are monotonically increasing or decreasing simultaneously on [a,b].
10. ### Generating inverse

Can someone help me find the inverse of $y = 7 + 5x^3 + x^7$? The fact that there is more than one variable to a unique power makes the inverse a little less obvious than usual.
11. ### Limit calculation

In the image below, how am I applying the limit laws wrong? I know the correct solution is 5, but I get -1! The problem I am facing is something related to my usage of the product rule for limits.
12. ### Calculating derivative using formal definition tricky

Can someone guide me in terms of finishing off the derivative computation that I have started below? Help is greatly appreciated!
13. ### Calculate derivative using formal definition

I'm really hoping someone can help me calculate the derivative (using the formal definition) for the following function: Here is my attempted solutions but I know it is wrong: Can someone tell me what I am doing incorrectly?
14. ### Using formal definition of a limit

Can someone help me complete the derivative computation below? I'm a bit stuck.
15. ### Series Convergence

What are the values of k that make the series convergent?
16. ### Calculating limit of rational function

For the question in the image, can someone please explain to me why we should interpret $\sqrt{x^2}$ as $|x|$?
17. ### Just need a little bit of help with these Calc questions-- integrals and volume

1) At a processing plant, over the course of an 8-hour day, workers move material into a pile, which is removed at a constant rate by a conveyor belt. The workers move material into the pile at rate 200e−0.5t units/hr while the conveyor belt removes the material at 50 units/hr. (a) Find the net...
18. ### Minimum volume of empty space in container with water

Volume of a container is $\frac{4 \pi}{3}$. Water can flow in and out of container.The volume of water in container is given by: $g(t), 0 \leq t \leq 4$, where $t$ is time in hours and $g(t)$ is measured in $m^3$. The rate of change of the volume of water in the container is: \$g'(t)= 0.9 - 2.5...
19. ### How to solve improper integral ln(x) / (x^2 + 1)

Hello, guys, I just want you to help me of finding the integral of \int_{0}^{1}\dfrac{\ln(x)}{x^2 + 1}\mathrm{d}x. I just know that is improper.
20. ### Calculus 3

Curve \mathcal{K} is defined as an intersection of surfaces z^2 = 2x^2 + y^2 and z = x + 1. The curve's projection on plane z = 0 is positive oriented. Parametrise the curve \mathcal{K}. Find the Frenet-Serret basis of the curve \mathcal{K} at point A(2, 1, 3). Find the...