AP BC CALC
Series
1. Geometric 2. Sum of geometric series = a / (1-r)
a. a = constant (same as number if n =1 is plugged in)
b. r = ration being multiplied by
3. Convergence
a. interval of convergence
i. a. |inside of parenthesis| < 1 (radius of convergence)
ii. b. then check endpoints to see if converges or diverges ther
b. 1/n does not convege
c. -1^n / k does converge (if n is odd)
Convergence tests
AP BC CALC
Fundamental theorems
1. first: if a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b] then
2. second: if f(X) continuous … then the derivative is
a. in other words: replace the t with the upper limit x
i. if x = 2x, then replace all t’s with 2x’s
ii. and multiply by the derivative of the upper limit
Rolle’s theorem
1.
f'(x) =
a. does not necessarily exist
i. ex. absolute value function
Limits
1. L’Hopitals rule
a.
i. if limit is not clear from function, then the function can be
differentiated to find the limit
ii. if first differentiation is too inconclusive then differentiated
again
2. sin and cos rules as limit approaches 0
a. sin x / x = 1
b. cosx-1 / x = 0
c. sin ax / x = a
d. sin ax / sin bx = a/b
3. for a function to be differentiable it needs to have a limit and be continuous
at a point
a. vertical tangent ≠ differentiable
b. horizontal tanget = differentiable
c. continuous ≠ hole
d. no limit if a sharp angle (Ex. corner of a trapezoid)
Taylor and Maclaurin series
1. Series (taylor) a. f(n): derivative degree
b. a: number that series is centered at
c. If f(a) = 0 then it is a maclaurin series
i. = (f(n)(0) x^2) / n!
d. First term of series is f(a)
AP BC CALC
i. Since:
1. 0! = 1
2. x0 = 1
2. lagrange error
a. error of an approximations is equal to the next degree in a taylor
polynomial, if the series is alternating b. if however the series is not alternating then one needs to find the
maximum error for the next value
i. find max value of f(n+1)(c) then plug in that number to x
1. remember to multiply by(X-a)N+1 and divide by n!
3. Know by heart
1. … xn
1. …xn / n!
1. (-1)n x2n+1 / (2n+1)!
1. (-1)n x2n / (2n)!
- if given one of these with a different value of x, you can replace all x’s with
the new x value
- if one of these is multiplied by a constant then each part can be multiplied
by a constant
Derivatives
1. In order to be differentiable the function must be continuous and have a limit
at that point
2. Definition
or
a. f ‘ (a) is the derivative
i. ex. tan (a+h) – tan a / h
1. f(a) = tan (a)
2. f ‘ (a) = sec2a
3. Graph
a. f(x) = graph
b. f ‘ (x) = slope
i. if 0 then may be a min or max
1. Have to test to check that it is
a. with first or second derivative test
AP BC CALC
i. first derivative test
1. max if d’ changes from + to – at
point
2. min if d’’ changes from – to +
ii. second derivative test
1. if d’’ > 0 then min
2. if d’’ < 0 then max
3. if d’’ = inconclusive
c. f ‘’ (x) = concavity
i. if 0 then may be a point of inflection
1. where concavity switches signs
4. Space
a. f(t) = position
i. max position d’ = 0
ii. if asked to find total distance travaled then its |speed|dt
1. absolute value because ewe care about when the
velocity was negative as well
b. f ’(t) = velocity
i. if asked to find speed and given dy/dt and dx/dt
1.
ii. max veloctity d’’’ = 0
iii. speed increasing if:
1. V is positive and A is positive
2. V is negative and A is negative
iv. Instantaneous velocity = dv/dt
v. average velocity = t(b)-t(a) / b-a
c. f ’’(t) = acceleration
5. Sin, cos, tan
a. cos  -sin
b. sin  cos
c. tan  sec2x
6. Quotient rule
a. u/v  (u’v – uv’)/v2
7. e
a. eax  aeax
8. ln(u)
a. du/u
9. Sin and cos
a. sin ax  a cos ax
b. cos ax  -a sin ax
c. sin (u) = cos(u) * du
d. cos (u) = -sin(u) * du
10. Tangent line
a. find the derivative
b. isolate y’
AP BC CALC
c.
d.
e.
f.
g.
plug in (x,y) values to get y’
y’ = slope
equation: y-y1 = r(x-x1)
if parametric then dy/dx still works
! when differentiating remember to differentiate y and by doing so a
adding a y’
11. d’ (ln(x))  1/x
12. Polar curves
a. Change to rectangular then differentiate
i. y = r sinϑ
ii. x = r cosϑ
b. to find slope of tangent line continue:
i. then dy/dϑ / dx/dϑ
ii. then plug in ϑ
13. h(x) = f(g(x)
a. h’(x) = f’(g(x))*g’(x)
b. h’’(x) = f’’(g(x))*[g’(x)]2 + f’(g(x))g*’’(x)
14. linear functions
a. d’ = 1
b. d’’ = 0
Area for polar curve
1.
Curve length
1. Rectangular coordinates -
2. parametric coordinates Integrals
1.  1/x = ln(x)
2.  du/u  ln |u| + C
a. ln (e) = 1
b. ln (1) = 0
c. ln (0) = dne
3. e
a. eax  eax / a
b. eudu  eu
i. if du needs a constant before then:
1. add constant inside integral and put the constants
reciprocal outside parenthesis
2. final answer is constant that is outside integral * eu
4. u du  u
AP BC CALC
a. if trig functions are used or functiosn to powers are used, only care
about base of function
i. ex. cos2x  ignore the second power
5. Integration by parts
a. when to use: if cant integrate and denominator can be factored
b. how:
i. Factor out the denominator
ii. Create new equation: (letter) A / newly factored part 1 +
B / newly factored part 2 …
iii. Then multiply out theƒ denominator
iv. Then make the new equation equal to original numerator
v. Solve for X’s in order get rid of as many factors so that the
letters can be given numerical values
1. Each time getting a new value for a, b or ….
vi. Then insert new value instead of a , b …
vii. Integrate
viii. New equations
6.  u*dv
a. If an integral has two parts one of which can be integrated and the
other can be differentiated then this is an easy way to integrate
b.  u*dv = u*v - du*v
7. integrals can be added and subtracted, but not multiplied and divided
Sums
1. trapezoidal approximation
a.
i. b = end point
ii. a = begging point
iii. n = number of intervals
1. only if intervals are equal
b. uneven intervals
i. base of each interval multiplied by (side 1 + side 2) / 2, for all
intervals
2. Riemann
a. left endpoint with equal intervals
i.
b. right endpoint with equal intervals
i.
c. midpoint
i.
d. not equal intervals
AP BC CALC
i. get rid of (b-a / n) then each yn value must be multipled by
subinterval length
1. ex. left enpoint (x1-x0)*y0
Euler’s method
1. yn = yn-1 + h*f(xn-1,yn-1)
a. y’ = f(x,y)
b. h – step size
2. n = 0 is the initial condition given
3. each step increases x by step number and y by the equation
Logistic differential equations
1. equation: kP(1-P/L)
a. L - limit
b. P – only variable
2. max rate of growth is L/2
a. growth rate
3. if k>L then the population is decreasing and d’ is negative
4. if k<L then the population is increasing and d’ is positive
5. limit occurs when the derivative equal zero
a. usually make the inside of the equation equal 0, thus finding the P
which is the limit
3D shapes and area between two cruves
1. area between two curves
a. top curve – bottom curve
2. 3D shapes, remember not to do diameter, rather the radius
3. volume of a solid generated
a. if two lines, first curve^2 – second curve^2
i. R12-R22 dx
4. if perpendicular to x axis in terms of y, then everything is in terms of y =
a. regular
5. if perpendicular to y axis then need to make everything in terms of x =
6. if creating 3D circle then its (1/2 *r)2
7.
Other
1. area of equilateral triangle = (s2√3)/ 4
2. a curve has a slope means first derivate is …
3. average rate of change
a.
4. if third variable, then you have x,y,x’, and y’
5. watch out for product rule in volume problems
AP BC CALC
Mean value theorem
1. If a function is continuous and differentiable for [a,b] there exists at least one
value c where the tangent is equivalent to the slope between the two lines
a. f(b)-f(a) / b-a = f’(c)
i. to find c
1. differentiate the equation and make that equal to the
slope of the line
Rolle’s theorem
1. if y = f(x) is differentiable at all points [a,b] and f(a) = f(b) then there is at
least one number c between a and b that f’(C) = 0
Min and max
1. check end points and critical points
2. use either first derivative test or second derivative test
a. 1st d test: if going + to – then max, if – to+ min
b. 2nd d test: if + then min, if – then max
Inverse functions
1. first find x value that when plugged in produces the correct x-value output
(which is the x value point that your supposed to find the inverse at)
a. or y value input …
2. second: differentiate function
3. third: get the reciprocal of the differentiated function
4. fourth: plug in x value from step one
a. ex. f(x) = x2 find a derivative of f-1(x) at x = 9
i. first: 32 = 9
ii. second: 2x
iii. third: 1/2x
iv. fourth: 1/ 2*3 = 1/6
5. (or just switch y and x and then integrate)
Inverse derivates
1. Sin-1 x = 1/√(1-x2)
2. cos-1 x = -1 / √(1-x2)
3. tan-1 x = 1 / 1+x2
4. cot n-1 x = - 1 / 1+x2
5. sec-1 x = 1 / |x|√(x2-1)
6. csc-1 x = -1 / |x|√(x2-1)
Remember
1. try to bring constants out whenever possible
2. ex+c = Cex
3. add constant (C) after integration
Alternating series converges if:
1. tn > 0
a. (the terms must alternate in signs)
2. tn > tn-1
3. tn  infinity as n  infinity