I have a question on the formulas for arc length and surface area.
Do you use the formula: s= \int_{c}^{d}\sqrt{1+[g'(y)]^2}dy only when you are provided with a function x=g(y)?? Can you convert that to y=g(x) and solve it by replacing g'(y) with y(x), changing the bounds and the dy to dx...
Homework Statement
Question regarding #16
Homework Equations
Riemman Sum
The Attempt at a Solution
I know that the limit of the Riemman Sum is basically the integral. However, I do not know where to go from there. Do I need to use the Summation formulas? Thanks
Homework Statement
I'd like hints as to how to solve this problem. Thanks!
\lim_{x\rightarrow{\inf}}{\frac{x^2-4}{2+x-4x^2}}
Homework Equations
I think I would begin by dividing out the fraction to get rid of the x^2? Is this the right way to start?
The Attempt at a Solution
Homework Statement
I'm having difficulty graphing with respect to y (unless I graph it point by point).
For example, I find it hard to graph: f(y)= \frac{y}{\sqrt{16-y^2}}
Does anyone know of an easy method to graph such a function or know how to do it on a Ti 84+ SE? Also, it would...
Writing "Cram Sheets" for Math/Science Classes...
I like to write "Cram Sheets" for Math/Science Class that include my common mistakes, important notes, and any extra information that my textbook does not accentuate or simplify. I like them to be neat and tidy (and hopefully digital so I can...
My book has a problem that requires you to separate variables (one side has all the y terms and one side has all of the x terms):
\sin{xy'}=\cosx
Equation after separation of variables:
dy=\cot{x}dx
My question is, how do you know that the y' is contained within the sine function or...
I do not understand the process of separating variables such as in derivatives. I thought that dy/dx is just the rate of change of y with respect to the independent variable x. Why can you physically move dx (like multiply it on both sides)?? What would "dy" be reffered to as then? Simply the...
Here are some questions I have concerning AP Calculus that I have compiled while doing my homework assignments. Please help me answer them. Thank you very much!
Questions:
-Do you never need to worry about the chain rule when integrating?
I'll add on to these questions when I have more.
Do you always need to watch out for "limitations?" How do you memorize them?
I have a question about "limitations" given within any theorem. For example: Rolle's Theorem states:
So the limitations would be: "Let f be continuous on the closed interval [a,b] and differentiable on the open...
I'm currently a high school junior taking a variety of AP courses. I'm getting A's with the exception of a few B's. I really want to conduct a science fair project, however, and I don't want to rush it (as long as I can finish before senior year).
Should I focus on maintaining good grades...
I've been thinking... Since derivatives can be written as:
f'(c)= \lim_{x\rightarrow{c}}\frac{f(x)-f(c)}{x-c}
and for the limit to exist, it's one sided limits must exist also right?
So if the one sided limits exist, and thus the limit as x approaches c (therefore the derivative at c)...
My dad just bought me Mathematica 5.1 and Mathcad 12 and I have no idea how I should use it, if ever. I'm currently in AP Calculus in high school so I don't know what features I should use that would enhance my learning and understand of math from algebra all the way up to Calculus. Any ideas or...
Homework Statement
\int \frac{1}{1+\sqrt{2x}}dx
Homework Equations
u=1+\sqrt{2x}
\sqrt{2x}=u-1
dx=(u-1)du
The Attempt at a Solution
I was able to get it down to:
\int (1-\frac{1}{u})du
= u-\ln{lul}}+C
= 1+\sqrt{2x}-\ln{l1+\sqrt{2x}l}+C
However, my book says that...
I don't know how I should study Calculus and math related subjects. Here's the routine I follow as of now: After I come back from school everyday after goingto Calculus class I read through the notes and do all example problems. Then I start and finish the assignment the next day.
However...
I don't get why we need to use differentials and why they are the way they are.
For example: dy=f'(x)dx vs. the derivative \frac{dy}{dx}=f'(x)
Why are they equivalent? Why are integrals written in the differential form? I don't get the purpose of it. (other than to be used as an...
Calculator program gives incorrect results (Definite Integrals/ Area)????
I've inserted this definite integral into my calculator program:
\int_{-1}^{2}x(x^2-4)dx
and my calculator gives me -9/4 for the integral, which is what my book's answer key has written down.
However, the area...
I'm confused here.. My definite integral doesn't match by Riemman Sum... and it should right? I think that I have not integrated correctly. Can someone help me spot the problem? Thanks.
Find the Area of the region bounded by:
f(x)=5-x^2 , [-2, 1]
Using the Riemma Sum idea (limit of the...
Homework Statement
With what intial velocity must an object be thrown upward (from ground level) to reach the top of the Washington Monument (approx 550ft)?
Homework Equations
Here's what I know:
s(0)=0
a(t)=-32ft/s^2
s(t_{max})=550\\
s'(t_{max})=0
The Attempt at a Solution...
Homework Statement
Find F'(x) if
F(x)=\int_{0}^{x^3}(\sin (t^2))dt
The Attempt at a Solution
Here's what I did:
F(x)= -\cos (t^2)\biggr]^{x^3}_{0}
and I get: F(x)= -\cos (x^6) +1
F'(x)= sin (x^6)(6x^5)
However, the book's answer is F'(x)= 3x^2 \sin(x^6)
How did...
[SOLVED] Definite Integrals
Homework Statement
\int_{1}^{3}x^{2}dx
Homework Equations
The Attempt at a Solution
Why is the answer 26/3? I got 4 by using the limit/Riemann Sum definition. Is this one method to calculate definite integrals?
I'm studying concavity and the second derivative test and it is starting to become more confusing because the second derivate is the slope of the slope of the original function. Do you need to become so familiar with this that you have a "picture" of the graphs in your head? I mean, one can just...
Does "f" mean the same thing as "f(x)"??
I'm wondering if f means the same thing as f(x). Does f refer more to the graph and f(x) refers to the y-coordinate of the graph?
Which one should I use to complete this sentence?:
___ is increasing on interval (-1,2)
or
___ has a relative...
Hello there! Please help me relieve my confusion. Thanks!
For \frac{d}{dx}[y^{3}] , why do you need to use the chain rule on this equation? Basically, the chain rule is used on almost every function right? It is just that we do not see the dx/dx since it equals one, for example...
I need to find the derivative of: f(t)=-2t^{2}+3t-6
However, I do not know if I'm writing it out correctly. Please tell me if I'm doing anything wrong, thanks!
My Solution:
\frac{d[-2t^{2}]}{dx}+\frac{[3t]}{dx}-\frac{d[6]}{dx}
Is that the correct way to write it out? Do you just...
I'm stuck on this limit function and I don't know what to do next. Please help me out. Thanks!
\lim_{x\rightarrow 0^{+}}(\frac{\csc2x}{x})
I just turned csc2x into 1\sin2x so then I have:
\lim_{x\rightarrow 0^{+}} (\frac{1}{x\sin2x})
Then I used the trig identity: sin2x=2sinxcosx...
I'm having a hard time find the limits of these trig functions. Please help me with it. Thanks in advance.
1. \lim_{x \rightarrow \pi}(\frac{\sqrt{x}}{csc x})
From this function I know that csc x= \frac{1}{sin x} which cannot equal 0.
X, therefore, cannot equal \pi n where n is any...
I'm wondering what the best way is to solve:
\lim_{x \rightarrow 3^{-}} \frac{x}{\sqrt{x^2-9}}
I'm pretty sure that f(x) is not equal to zero but I can't seem to manipulate it to cancel out (x-3). Also, when solving these types of problems, can you use the same rules as regular limits and...