Ms. Matroni,
On behalf of the AP Calculus Class of 2010, I'd like to thank you for the year. It's been great an insanely fun. I think it goes without saying that everyone will miss this class and the randomness that ocurred. Pity this blog wasn't used for anything more, and I'm surprised no one else has posted sine the day of the AP exam...Anyways, it's been a blast.
Sunday, May 23, 2010
Tuesday, March 16, 2010
Answers to Area/Volume Worksheet
The answers to today's worksheet can be found on Xavier's AP Calc website. It was just easier to type them out using MS Word, rather than attempt those symbols on here. :)
Go check them out and make sure you understand each answer.
If you have any questions, post them here or email me.
Good luck studying!
Go check them out and make sure you understand each answer.
If you have any questions, post them here or email me.
Good luck studying!
Friday, February 26, 2010
Sunday, January 31, 2010
Wednesday, January 27, 2010
Friday, January 22, 2010
January 22 Homework
Just fyi - if you're having trouble finding the website I mentioned in class in order to download the question you're supposed to answer, I posted the link on the Xavier website that will lead you directly to the page. Then just click the link for the 2003 (NOT FORM B) exam and have fun!! :)
Thursday, January 14, 2010
MY Notes cause i am awesome
So this is what she has on her website about things that we have done in class, im going to write any notes I have on the sections, Integration stuff is too tough to type but Rob Furatero put that on the blog as well as derivative rules, Sorry for the jokes and if u don’t understand something just write a comment or something, Ms. Matroni is in learning center from 8-12 tommorow and is lonely
Limits-from a chart: just if the numbers are approaching a number on both sides of the number-from a graph: just know that + is from the right and – is from the left once again figure it out, no big holes that’s really only rules,
Fail to exist if: behaviors differ on the two sides, unbounded behavior, and oscilating behavior (lie detector test when Sypa claims he isn’t the father)-rules at infinity: 1) degree of numerator <> degree of denominator, THE LIMIT DOES NOT EXIST 3) if they are equal it’s the coefficient of the highest degree of the numerator/ coefficient of highest degree of denominator -analytically -direct substitution: plugging in the number, if u get 0/0 -dividing out technique: factor out and see if u can cancel out, then try plugging in again -rationalizing technique: if u got square roots rationalize by conjugal pairs (Father O Hare) -L'Hopital's rule: if u get indeterminate form, take the derivative of the numerator and the denominator than try again, if again indeterminate, keep going Continuity-3 criteria for continuity 1) F(a) has to exist, f must be defined at X=a 2) lim. Of Xàa has to exist 3) 1 and 2 have to be the same value-removable/non-removable discontinuities: a hole Is removable and giant gaping hole is nonremovable, the difference is that a hole can be factored out of denominator while giant gaping hole cannot be.
Intermediate Value Theorem (not in her list but oh well): if it is continuous on the closed interval [a,b] and k Is any number between f(a) and f(b) then at least one number c in [a,b], such that f(c)=k (haha that looks like a bad word) Derivatives-differentiability (when does a derivative not exist?): at a corner, at a cusp, vertical tangent and at a discontinuity -differentiability implies continuity (but not the other way around): that’s really it I guess -the derivative as a "special" limit-rules - basic rules, product rule, quotient rule, chain rule, trig functions, inverse trig functions, exponential functions, natural log functions (ON BLOG) -implicit differentiation (when xs and ys are derived): 1) treat x and y as variables and take the derivative as you normally would 2) after taking the derivative of any y multiply by y^1 3) gather all terms with y^1 to one side 4) solve for y^1Applications of Derivatives-slope of tangent line (normal line) F^1 (x)= f(x+h)- F(x)/h where h is what the limit is approaching (IDK why we need to know that)-instantaneous rate of change: that’s when u have a graph and u have to know the slope at a specific point but the slope keeps changing, USE THE DERIVATIVE-physics applications (distance, velocity, acceleration): S(t) is a function representing an object’s position, the first derivative determines velocity while the 2nd derivative determines acc. -the mean value theorem: theres a giant explanation but all it really means is that when the line is differentiable, then the average value of the line is equal to the instantaneous value of the line somewhere, just accept that-related rates word problems: 1) identify all known and unknown variables/ rates of change 2) identify an equation that relates your variables together 3) take derivative implicitly (that’s when u have the y^1s) 4) Do as Sypa and Plug and Chug-extreme value theorem (absolute max and min): If f is continuous on a closed interval than f has both an abs. min and max on the interval, that’s why when determining abs. mins and maxs on a graph u have to test all rel. min or maxs and then ur two points of the interval-curve sketching -first derivative charts (where f(x) has relative max and min, increasing, decreasing): 1) if f^1 > 0 then f is increasing if f^1 <0>0 then concave up, f^ll < 0 concave down, f^ll=0, pts. of interest as long as its changing concavity-using the graph of the derivative to find properties of the original function, yeah so um, yeah when u have a zero on either graph of F^l or F^ll, plug into original problem so u can plot the point, also plug 0 in for x so u can find the y intercept, and plug in 0 for y to find x intercept if possible-optimization problems ("real world" max/min word problems): 1) identify the given quantities and quantities to be found 2) make a drawing 3) find an equation for the quantity to be maximized or minimized 4) find a second equation to plug into the first (all in terms of one equation) *****make sure u check that u actually solved for a max or the min by making sure that it increases to the left and decreases on the right for maximization and vice versa for minimization -linearization (tangents lines to approximate values): 1) look for a value that is easy to use (like when working with square roots look for a perfect square close by), then derive and plug in ur new x, to find the slope, next use that slope as well as ur x and y to find out the y intercept aka b in y=mx+b, lastly plug in your original x that crazy number u couldn’t figure out the answer to on ur own and plug into ur new equation and then solve for y, ur new answer should be close to the actual answer. (Sorry I understand that’s confusing if u have questions on that just message me or something) Integration-approximating area under a curve using rectangle and trapezoids: RAM is divided into 3 ways Left ram, middle ram and right ram, LRAM and RRAM are easy, for LRAM= (b-a/n)(f(x) the first y point, and then until one before the last point, for RRAM it is the same thing but you do not use the first point, the trick is how u draw ur squares are u drawing the flat line for the top from the left or from the right
TRAPEZOIDS: this is where it gets tricky everyone knows that the area of a triangle is A=1/2 h(b1+b2) okay that’s great, now to use this to find the area of a trapezoid under a curve we are making multiple trapezoids(the trapezoids are on their side) where the height is always the change in x so (b-a/n)*n is the number of triangles* again but since theres the one half in front it becomes (b-a)/2n, okay now each base expect for the first and the last is used twice because it is used in the triangle on the left as well as the triangle to its right, this is why we multiply by 2 every f(x) that is not the first or the last value of f(x) in all the equation for the area of a trapezoid under a curve is
A trap= (b-a)/2n multiplied by (f(1stx) +2f(2ndX) +…… +f(last X))
-solving definite integrals (exact area under a curve) (ON THE BLOG) Rob Furatero did it!!
Limits-from a chart: just if the numbers are approaching a number on both sides of the number-from a graph: just know that + is from the right and – is from the left once again figure it out, no big holes that’s really only rules,
Fail to exist if: behaviors differ on the two sides, unbounded behavior, and oscilating behavior (lie detector test when Sypa claims he isn’t the father)-rules at infinity: 1) degree of numerator <> degree of denominator, THE LIMIT DOES NOT EXIST 3) if they are equal it’s the coefficient of the highest degree of the numerator/ coefficient of highest degree of denominator -analytically -direct substitution: plugging in the number, if u get 0/0 -dividing out technique: factor out and see if u can cancel out, then try plugging in again -rationalizing technique: if u got square roots rationalize by conjugal pairs (Father O Hare) -L'Hopital's rule: if u get indeterminate form, take the derivative of the numerator and the denominator than try again, if again indeterminate, keep going Continuity-3 criteria for continuity 1) F(a) has to exist, f must be defined at X=a 2) lim. Of Xàa has to exist 3) 1 and 2 have to be the same value-removable/non-removable discontinuities: a hole Is removable and giant gaping hole is nonremovable, the difference is that a hole can be factored out of denominator while giant gaping hole cannot be.
Intermediate Value Theorem (not in her list but oh well): if it is continuous on the closed interval [a,b] and k Is any number between f(a) and f(b) then at least one number c in [a,b], such that f(c)=k (haha that looks like a bad word) Derivatives-differentiability (when does a derivative not exist?): at a corner, at a cusp, vertical tangent and at a discontinuity -differentiability implies continuity (but not the other way around): that’s really it I guess -the derivative as a "special" limit-rules - basic rules, product rule, quotient rule, chain rule, trig functions, inverse trig functions, exponential functions, natural log functions (ON BLOG) -implicit differentiation (when xs and ys are derived): 1) treat x and y as variables and take the derivative as you normally would 2) after taking the derivative of any y multiply by y^1 3) gather all terms with y^1 to one side 4) solve for y^1Applications of Derivatives-slope of tangent line (normal line) F^1 (x)= f(x+h)- F(x)/h where h is what the limit is approaching (IDK why we need to know that)-instantaneous rate of change: that’s when u have a graph and u have to know the slope at a specific point but the slope keeps changing, USE THE DERIVATIVE-physics applications (distance, velocity, acceleration): S(t) is a function representing an object’s position, the first derivative determines velocity while the 2nd derivative determines acc. -the mean value theorem: theres a giant explanation but all it really means is that when the line is differentiable, then the average value of the line is equal to the instantaneous value of the line somewhere, just accept that-related rates word problems: 1) identify all known and unknown variables/ rates of change 2) identify an equation that relates your variables together 3) take derivative implicitly (that’s when u have the y^1s) 4) Do as Sypa and Plug and Chug-extreme value theorem (absolute max and min): If f is continuous on a closed interval than f has both an abs. min and max on the interval, that’s why when determining abs. mins and maxs on a graph u have to test all rel. min or maxs and then ur two points of the interval-curve sketching -first derivative charts (where f(x) has relative max and min, increasing, decreasing): 1) if f^1 > 0 then f is increasing if f^1 <0>0 then concave up, f^ll < 0 concave down, f^ll=0, pts. of interest as long as its changing concavity-using the graph of the derivative to find properties of the original function, yeah so um, yeah when u have a zero on either graph of F^l or F^ll, plug into original problem so u can plot the point, also plug 0 in for x so u can find the y intercept, and plug in 0 for y to find x intercept if possible-optimization problems ("real world" max/min word problems): 1) identify the given quantities and quantities to be found 2) make a drawing 3) find an equation for the quantity to be maximized or minimized 4) find a second equation to plug into the first (all in terms of one equation) *****make sure u check that u actually solved for a max or the min by making sure that it increases to the left and decreases on the right for maximization and vice versa for minimization -linearization (tangents lines to approximate values): 1) look for a value that is easy to use (like when working with square roots look for a perfect square close by), then derive and plug in ur new x, to find the slope, next use that slope as well as ur x and y to find out the y intercept aka b in y=mx+b, lastly plug in your original x that crazy number u couldn’t figure out the answer to on ur own and plug into ur new equation and then solve for y, ur new answer should be close to the actual answer. (Sorry I understand that’s confusing if u have questions on that just message me or something) Integration-approximating area under a curve using rectangle and trapezoids: RAM is divided into 3 ways Left ram, middle ram and right ram, LRAM and RRAM are easy, for LRAM= (b-a/n)(f(x) the first y point, and then until one before the last point, for RRAM it is the same thing but you do not use the first point, the trick is how u draw ur squares are u drawing the flat line for the top from the left or from the right
TRAPEZOIDS: this is where it gets tricky everyone knows that the area of a triangle is A=1/2 h(b1+b2) okay that’s great, now to use this to find the area of a trapezoid under a curve we are making multiple trapezoids(the trapezoids are on their side) where the height is always the change in x so (b-a/n)*n is the number of triangles* again but since theres the one half in front it becomes (b-a)/2n, okay now each base expect for the first and the last is used twice because it is used in the triangle on the left as well as the triangle to its right, this is why we multiply by 2 every f(x) that is not the first or the last value of f(x) in all the equation for the area of a trapezoid under a curve is
A trap= (b-a)/2n multiplied by (f(1stx) +2f(2ndX) +…… +f(last X))
-solving definite integrals (exact area under a curve) (ON THE BLOG) Rob Furatero did it!!
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