<p>
[quote]
...remember, the derivative of the integral of f(x) from c to x (where c is a constant) is just f(x). </p>
<p>But, if it is the derivative of the integral of f(x) from c to 3x^2, you have to use the chain rule, so it would be 6x*f(x)
[/quote]
But how does that get the answer: F'(a)?</p>
<p>if anyone needs any BC help, IM me at Amol Is the Man. I'm no expert at calculus, but i think together we can work it out. Plus, it'd be a good review for me too--they say the best way to learn is to teach.</p>
<p>
[quote]
Q. lim(x → ∞) 4x*sin(1/x) = . . .</p>
<p>A. lim(x-> inf) sin(1/x)/ 1/4x -> (cos(1/x) * -1/x^2 )/ -1/4x^2 = 4cos(1/x) = 4
lim (x->inf) 4(cos(1/x)) = 4(cos0) = 4
[/quote]
</p>
<p>How did you go from 4xsin(1/x) to sin(1/x)/(1/4x)?</p>
<p>He just made the function in a fraction form: 4x * sin(1/x), sin(1/x) / (4x)^-1 [he just put 4x to the bottom side of the large fraction, but kept it in a form that keeps the function the same]</p>
<p>Sorry if my explanation does not make much sense, I am tired today...</p>
<p>How do you do the Lagrange stuff? That's the one thing with series that I am absolutely clueless on! Please write it in easy to understand terms (that's one reason why I don't get it, since I don't really understand the symbolic terms).</p>
<p>I think my prep. book explains it quite well. I'm going to quote it directly. </p>
<p>
[quote=PR AP Calculus AB and BC, ed. 2002-2003]
Sometimes you will be asked to find the "error bound" of a Taylor polynomial. This is also sometimes referred to as the "Lagrange error." The proof for the rule is quite difficult, so we will simply tell you the rule. If you are finding an nth degree Taylor polynomial, the error is the n+1th term. For example, if we wanted to find the error in the above example [Use a third degree Taylor polynomial to estimate e^0.2], we know that it's less than the fourth term evaluated at 0.2, which is (0.2)^4/4! = 0.000066...
[/quote]
</p>
<p>@waffle</p>
<p>Basically, the Lagrange bound error states that the max error of a partial sum of a series is the first omitted term at some c between 0 and a.</p>
<p>Example: If you want to approx e^0.2 to a max error of 0.0001, the first omitted term must be less than 0.0001:</p>
<p>e^0.2 approx = 1 + 0.2 + 0.2^2/2! + 0.2^3/3!</p>
<p>The first omitted term is 0.2^4/4!, which is the max error, which turns out to be 0.000067</p>
<p>Thats a funny coincidence, we both used e^0.2 as an example O_O.</p>
<p>Newton's Law of Cooling anyone?</p>
<p>That basically states that an objects cools/heats at a rate proportional to the difference between the room temp and the temp of the object, usually given by this diff equation: dT/dt = k(T - R) where R = room temperature, T = temp of the object, and k is any arbitrary constant.</p>
<p>sometimes the notation of (and I think I'm getting this wrong since I'm doing it from memory) the lagrange error bound will ask for something like the the parameter that is less than the absolute value of f(x) - P(x) -- is this just a way of asking for the error to be approximated or do you actually have to solve something inside of that absolute value?</p>
<p>oh, and what about problems with carrying capacity as used in a differential equations? what do you do with those besides identifying the carrying capacity and properties of it like as t approaches infinity, the carrying capacity is reached?</p>
<p>That is just a fancy way of saying that the error should be less than the parameter. You will never have to solve the stuff in that abs value, don't worry. From my experience, College Board rarely asks you to do very tedious work.</p>
<p>Keep posting problems!</p>
<p>miken...: NEWTON'S LAW OF COOLING</p>
<p>dy/dt= k(L-y)</p>
<p>therefore, y = L-(L-y0)e^-kt</p>
<p>L is the maximum</p>
<p>In order to get the equation of y (which shouldn't be asked on the exam), just separate the variable and take the integrals.</p>
<p>EVERYBODY READ THIS:</p>
<p>There is NO:
Simpson's Rule Approximation</p>
<p>KNOW, though:
Average rate of change- f(b)-f(a)/b-a
Definition of derivative (limit as h goes to 0 or x goes to a)
Average Value- integral over b-a
REMEMBER that f"(a)=0 DOES NOT imply that a is an infleciton point.
2nd derivative test- shows local max and mins- positive is concave up, negative is concave down</p>
<p>Fundamental Theorem- integral of f(x) is equal to F(b)-F(a)
-also the derivative of the integral of f(x) is just f(x)
Trapezoidal Approximation- change in x(y/2+y1+y2+...yn-1+yn/2)
Exponential Growth Rate- dy/dt= ky y=Ce^kt
Logistic Growth- dy/dt = K(L-y) or ky(1-(y/L)) y= L/(1+Ce^-KLT)</p>
<p>Euler's Method- f(a) + f'(x0,y0)(x-a)
L'hopital's rule- if limit is infinity over infinity or 0 over 0, take the derivative of the top and then the bottom of the fraction to find the limit.</p>
<p>What usually screws me over are polar coordinates. It is hard for me to think in terms of the polar plane...</p>
<p>Geometric Series- y=ar^n sum= a/(1-r)
Know Taylor series for sinx, cosx, Arctanx, and e^x
Harmonic Series- DIVERGES
Alternating Harmonic Series- CONVERGES
Alternating Series Test- alternating and decreasing- CONVERGES
Ratio Test- lim as n goes to infinity of the absolute value of a(n+1) over a(n)
DON'T DISTRIBUTE- I JUST COULDN'T DO SUBSCRIPTS</p>
<p>Derivatives- implicit, parametric (d2y/dx2), polar (the impossible one)
Lengths of Curves- functions, parametric, polar
Polar Area</p>
<p>THERE IS NO SURFACE AREA OF ROTATED SOLIDS (FUNCTIONS, PARAMETRIC, OR POLAR)!!!!!</p>
<p>yatta!- just remember that x=rcos(theta) and y=rsin(theta)
tan(theta)= y/x (use this to find theta for polar) and x^2+y^2=r^2</p>
<p>The area is 1/2(the integral from alpha to beta) of f(x)^2 or r^2</p>
<p>REMEMBER!!!!! Don't square r^2 again by accident to find the area. Most times you will find r^2 and just substitute in. Some people accidentally do it again.</p>
<p>This really is helping me, because I am studying too. I can't believe it's in one day. AAAAHHHH!</p>
<p>Thanks, Musikalgeak. I have a tendency to mess up with the integrand (like squaring it again, forgetting to square r at all, etc...)</p>
<p>Great tips.</p>
<p>Know Taylor series for sinx, cosx, Arctanx, and e^x</p>
<p>Can someone list those? Doesn't this depend on where the center is, or is this not being evaluated?</p>