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Stack #4633819
| Question | Answer |
|---|---|
| What is the formula for integrating a single variable function (for example x^2) | x^(exponent + 1)/(exponent plus 1) |
| What is a differential equation? | An equation where a d over d statement (ex dy/dx) is set equal to a function or constant. |
| How do you solve a differential equation? | multiple both sides by the denominator of the d statement. Then INTEGRATE |
| What happens to dx when you integrate? | it just goes away. lowkey just ignore it. |
| How can you integrate multiple single variable functions added/subtracted together? | "Integrate by parts". Just do them all seperately and add/subtract together. |
| What needs to be added to the end of any implicity integration? | + c :) |
| How do you integrate when x is in the denominator? | Translate to the x into a negative exponent. |
| If you are integrating two functions together (ex: 2x(x+5)), what do you do? | Distribute to make them into added/subtracted quantities, and solve. |
| If you are integrating a function with mutliple parts over a denominator ((x+1)/x), what can you do? | Split the fraction into two fractions (i.e x/x + 1/x) and integrate seperately. |
| What is the integral of sinx? | -cosx + c |
| What is the integral of cosx? | sinx + c |
| What is the intergral sec^2(x) ? | tanx +c (squared somehow one number) |
| What is the integral csc^2 (x)? | -cotx + c (squared somehow one number) |
| What is the integral of secxtanx | secx + c (sec alwayssss) |
| What is the integral of e^x? | e^x + c |
| What is the integral a^x (where a is a constant) ? | a^x/(ln(a) + c |
| What is the integral of (1/x) | ln I x I + c |
| What are the pythagorean identities? | cos^2 + sin^2 = 1 1 + tan^2= sec^2 1 + cot^2 = csc^2. |
| How can you use an initial condition to find what c is? | Plug in the point into the integrated equation and solve for c. |
| Explain the processes of sigma notation? | bottom--> (i = number to start on). top ---> number to stop at. If there a constant before a function it can be broight outside the notation, If two functions added/subtracted you can split the sigma. |
| whats the sigma of a constant times n | cn |
| Whats the sigma of i to n? | n(n+1)/2 |
| Whats the sigma of i^2 to n? | n(n+1)(2n+1)/6 |
| Whats the sigma of i^3 to n? | n^2(n+1)^2 / 4 |
| How can you a given sigma forumula to find the og sigm equation? | whatever number changes just put in an |
| What is the LOWER SUM? | Underestimate of the area under the curve. (when graph is CC up its an overestimate) |
| What is the UPPER SUM? | Overestimate of the area under the cuve. (whn graph is CC up its an overestimate) |
| How can you estimate the area under the curve? | (b - a)/ n (where b is the final endpoint, a is the first one, and n is the amount of rectangles/subintervals u want). TIMES the sigma notation of all f(point where a subinterval is) . LEFT hand means exclude last point. RIGHT hand means exclude first. |
| How can you estimate the area under the curve using midpoint? | Just do the same thing as a regular estimate but use the midpoint of the points of the subinterval |
| How can you calculate the exact area under the curve (fawk)? | (b -a)/n (the width) TIMESS f ( a + width x i) all in a sigma notiation where i equals 1 and it goes to n. Then use that answer as the limit approaches infinity. (the f part is plugged into x for the equation and that is timed by the width) |
| How can you simply the fawk equation? | Sigma by parts and always pull out constants and the n. Then use summation formulas. |
| How can you used the simplified fawk to actually find the exact area? | use the final equation ahd evaluate it as a limit set to infinity . |
| What are the parameters for fawk to work? | Rectangles must be even in size. Continuous. FAWKKKKK |
| What are Riemann Sums? | Finding exact area under curve using uneven widths :P |
| What is the formula for the Riemann Sum? | Sigma of heights times the width of all parts (added together) |
| When is area negative??? | if area is under the line. or if a is bigger than b. |
| When is area positive??? | Above the curve, or when it is below the curve BUTTTTT b is bigger than a (b is the top number of sigma) |
| What is the formula for area of a trapezoid? | (b1 + b2)/2 TIMES height |
| How can you switch the bounds of a integral? | make outside negative :) |
| How can u find the integral of a point inbetween a and b? | (low a and high c integral) + (low c high b integral) were c is the point u want |
| What is the formula for the trapezoidal rule (to find area under curve)> | 1/2 (b - a)/n times (f of first point plus f of all the other points times 2 plus the end point NOT times 2). |
| What is the fundamental therom of calculus?? | definite integral a to b = integral of b - integral of a |
| Whats the mean value for integrals? | definite integral of b to a = (b - a) f(c) -----> definite integral / (b -a) to get AVERAGE. Plug that into the y spot into og equation and solve for x. |
| What is the 2nd fundamental theorem of calculus? | |
| What is the formula for area of a trapezoid? | (b1 + b2)/2 TIMES height |
| How can you switch the bounds of a integral? | make outside negative :) |
| How can u find the integral of a point inbetween a and b? | (low a and high c integral) + (low c high b integral) were c is the point u want |
| What is the formula for the trapezoidal rule (to find area under curve)> | 1/2 (b - a)/n times (f of first point plus f of all the other points times 2 plus the end point NOT times 2). |
| What is the fundamental therom of calculus?? | definite integral a to b = integral of b - integral of a |
| Whats the mean value for integrals? | definite integral of b to a = (b - a) f(c) -----> definite integral / (b -a) to get AVERAGE. Plug that into the y spot into og equation and solve for x. |
| What is the 2nd fundamental theorem of calculus? | If top bound if variable and matches integrated eqation and youre taking the derivative, can get rid of integral (just use chain rule). |
| Describe how to find aspects of a F(x) = integral of f(x) | F( of a point) = definite integral of first point bound lower from eqn and upper bound is point you want. F(x) is increasing--F'(x) = f(x) > 0 (line above line) is F(ppt) pos? --Do def integral up to that point. H(x) max --> H'(x) = f(x) = 0/DNE |
| Inverse trigonomic functions? | du/sqr(a^2 - u^2) = arc(u/a) + c du/sqr(a^2 + u^2) = 1/a arctan(u/a) + c du/usqr(u^2 - a^2)= 1/a arcsec IuI/a + c |
Created by:
razawisza