Calculus II final review Questions and Answers
Sinh(x) *** (e^x-e^-x)/2
Cosh(x) *** (e^x+e^-x)/2
How do you eliminate the parameter? *** Solve for t
Pythagorean Identity *** cos^2x+sin^2x=1
Speed *** √(dx/dt)^2+(dy/dt)^2
Acceleration *** √(d^2x/dt^2)^2+(d^2y/dt^2)^2
Second derivative on a parametric curve *** (d/dt * dy/dx)/(dx/dt)
Integration by parts *** ∫ u dv = uv - ∫ v du
L.I.A.T.E *** logarithmic, inverse trig, algebraic, trig, exponential
Partial Fractions *** An/(x-an)...
√a^2-x^2 *** x=asin(theta)
a^2+x^2 *** x=atan(theta)
Right sum *** Delta x * Σ from j=1 to N f(a+j* Delta x)
Left sum *** Delta x * Σ from j=0 to N-1 f(a+j* Delta x)
Trapezoidal Sum *** Delta x [(f(x0)+f(xN))/2] + Σ from j=1 to N-1 of
f(a+j*delta x) OR 1/2(Right sum + Left Sum)
Midpoint/Tangent sum *** Delta X Σ from J=1 to N of f(a+(j-1/2)Delta x)
, Simpson's rule *** Trap sum/3 + 2*Mid Sum/3
Simpson's rule (Without using the sums) *** 1/3*delta x[y0+4y1+2y2 ... + 4yn-
1+yn] (Starts by adding the first Y and then alternates with 2 &4 but N-1 must
always end with 4)
Error bound for Simpson's rule *** [Ksub4*(b-a)^5]/180* N^4 where K sub 4 =
the max value at the fourth derivative of the function within the given interval
P integral for improper integrals *** if P is <=1 Diverges if P is > 1 converges to
(a^(1-p)/p-1)
Integral comparison test comparing f(x) to g(x) *** if g(x)> f(x) and g(x)
converges f(x) converges
if g(x)< f(x) and g(x) diverges f(x) diverges
if g(x)> f(x) and g(x) diverges f(x) is unknown (try again)
if g(x)< f(x) and g(x) converges f(x) is unknown (try again)
Finding volume of Revolution using Disk method *** ∫ from a to b of π*[f(x)]^2
dx. Use when rotation creates a disk who's radius r =f(x ) so that the integral is just
πr^2dx
Finding volume of Revolution using Washer method *** ∫ from a to b of
π[(R(x))^2-(r(x))^2]dx. Use when rotation creates 2 disks or a "hole" and then
subtract big R from little r
Finding volume or Revolution using Cylindrical Shell method *** ∫ from a to b
of 2πx*f(x) dx. When rotated around the y axis and bounded by x
Arc Length over [a,b] *** ∫ from a to b √1+[f'(x)]^2 dx
Area of a Surface of Revolution *** 2 π ∫ f(x) √1+[f'(x)]^2 dx
Arc Length of a Parametric curve *** ∫ √x'(t)^2+y'(t)^2 dt from a to b
Sinh(x) *** (e^x-e^-x)/2
Cosh(x) *** (e^x+e^-x)/2
How do you eliminate the parameter? *** Solve for t
Pythagorean Identity *** cos^2x+sin^2x=1
Speed *** √(dx/dt)^2+(dy/dt)^2
Acceleration *** √(d^2x/dt^2)^2+(d^2y/dt^2)^2
Second derivative on a parametric curve *** (d/dt * dy/dx)/(dx/dt)
Integration by parts *** ∫ u dv = uv - ∫ v du
L.I.A.T.E *** logarithmic, inverse trig, algebraic, trig, exponential
Partial Fractions *** An/(x-an)...
√a^2-x^2 *** x=asin(theta)
a^2+x^2 *** x=atan(theta)
Right sum *** Delta x * Σ from j=1 to N f(a+j* Delta x)
Left sum *** Delta x * Σ from j=0 to N-1 f(a+j* Delta x)
Trapezoidal Sum *** Delta x [(f(x0)+f(xN))/2] + Σ from j=1 to N-1 of
f(a+j*delta x) OR 1/2(Right sum + Left Sum)
Midpoint/Tangent sum *** Delta X Σ from J=1 to N of f(a+(j-1/2)Delta x)
, Simpson's rule *** Trap sum/3 + 2*Mid Sum/3
Simpson's rule (Without using the sums) *** 1/3*delta x[y0+4y1+2y2 ... + 4yn-
1+yn] (Starts by adding the first Y and then alternates with 2 &4 but N-1 must
always end with 4)
Error bound for Simpson's rule *** [Ksub4*(b-a)^5]/180* N^4 where K sub 4 =
the max value at the fourth derivative of the function within the given interval
P integral for improper integrals *** if P is <=1 Diverges if P is > 1 converges to
(a^(1-p)/p-1)
Integral comparison test comparing f(x) to g(x) *** if g(x)> f(x) and g(x)
converges f(x) converges
if g(x)< f(x) and g(x) diverges f(x) diverges
if g(x)> f(x) and g(x) diverges f(x) is unknown (try again)
if g(x)< f(x) and g(x) converges f(x) is unknown (try again)
Finding volume of Revolution using Disk method *** ∫ from a to b of π*[f(x)]^2
dx. Use when rotation creates a disk who's radius r =f(x ) so that the integral is just
πr^2dx
Finding volume of Revolution using Washer method *** ∫ from a to b of
π[(R(x))^2-(r(x))^2]dx. Use when rotation creates 2 disks or a "hole" and then
subtract big R from little r
Finding volume or Revolution using Cylindrical Shell method *** ∫ from a to b
of 2πx*f(x) dx. When rotated around the y axis and bounded by x
Arc Length over [a,b] *** ∫ from a to b √1+[f'(x)]^2 dx
Area of a Surface of Revolution *** 2 π ∫ f(x) √1+[f'(x)]^2 dx
Arc Length of a Parametric curve *** ∫ √x'(t)^2+y'(t)^2 dt from a to b
