Calc BC AP Test Review

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  1. Volume of a Solid with Cross Section

    Volume of a Solid with Cross Section

    • V=ʃab A(x)dx
    • (cross section is perp. to height of solid)
  2. Euler's Method when y'=F(x,y)

    Euler's Method when y'=F(x,y)

    • x1= x0+h y1= y0+hF(x0,y0)
    • h is step size
    • (smaller step size, better approx.)
  3. Disc Method about the line h

    Disc Method about the line h

    • V= (pi) ʃab( f(x) - h )2 dx,
    • where | f(x)-h |=radius
    • (if x-axis, h=0)
  4. Washer Method between f(x) and g(x)

    Washer Method between f(x) and g(x)

    • V=(pi) ʃab [ (f(x))2 - (g(x))2 ]dx
    • Outer radius - smaller radius
  5. What is the Squeeze Therom?
    if g(x)< f(x) < h(x) and Limx->a[g(x)= h(x)]=L, then so does that same limit of f(x)
  6. What is the Intermediate Value Therom?
    If a function f is continuous on a closed interval [a,b] and k is a number with f(a) < k < f(b), there exists a number inside the interval where f(c)=k
  7. What is...

    d/dx (au) =
    d/dx (au) = au ln(a) u'
  8. What is...

    d/dx log(u)=
    d/dx log(u)= 1/(u lna) * u'
  9. What is Rolle's Therom?
    • Rolle's Therom: if f is continuous [a,b] and differentiable (a,b), and f(a)=f(b)=0
    • then there is a number c that exists in (a,b) such that f'(c)=0
    • *rel. mins and maxs
  10. What is the Mean Value Therom?
    • Mean Value Therom: if f is continuous [a,b] and differentiable (a,b) thn there exists a number c in (a,b) that f'(c)=(f(b)-f(a))/(b-a)
    • *secant line slope passing through points a and b are equal to tangent slope of c
  11. General Procedure for Related Rates Problems!!
    • 1.Read problem, draw diagram
    • 2.Variables and Math symbols for given info
    • 3. Write and equation involving the rate of change to be determined(reduced to one variable)
    • 4.Differentiate with respect to time
    • 5.Plug in values and solve for desired rate
    • 6.Answer with units of measure
  12. General Procedure for Solving Applied Max and Min Problems!
    • 1. Write Primary Equation and Secondary Equation
    • 2. Solve so Primary Equation is in terms of one variable
    • 3. Find feasable domain
    • 4. Find critical values (f' = 0) of the function on the feasable domain
    • 5. Test critical numbers and endpoints
    • 6. Answer in context of the problem
  13. What is the Average Value of a Function? from (a,b)
    Average Value of a Function: 1/(b-a) ʃab f(x)dx
  14. Area of an equilateral triangle

    A=(s2 31/2) / 4
  15. Position Function
    Instantaneous Velocity
    Instantaneous Speed
    • Position Function = s(t)
    • Instantaneous Velocity = s'(t) = v(t)
    • Acceleration = s''(t) = v'(t) = a(t)
    • Instantaneous Speed = |v(t)|
  16. First Fundamental Theorem of Calculus
    ʃab f(x)dx = F(b) - F(a)
  17. Second Fundamental Theorem of Calculus
    F(x)= ʃab f(t)= f(x) , F'(x)=f(x) at every point of [a,b]
  18. Area between two curves f(x) and g(x)
    • ʃab [f(x)-g(x)]dx
    • upper curve - lower curve
  19. Distance Trveled Problems!!

    Displacement from t1 to t2
    • Displacement from t1 to t2 = ʃt1t2 v(t)
    • Total distance traveled is the same thing but the absolute value of v(t)
  20. p-series
    • is a series of the form, 1+1/2p=1/3p=1/4p+...+1/np+...= sum( 1/np, n, 1, infinity)
    • Series converges when p>1
    • Series diverges when 0<p<1
  21. harmonic series
    special p series, where p=1, so the series diverges
  22. geometric series
    • sum(ar(n),n,0,infinity).
    • series converges when |r|<1
    • sum of the series is a/(1-r)
  23. integral test
    if an=f(n) where f is continuous, positive, decreasing funciton on [c,infinity)

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Calc BC AP Test Review
2011-04-27 01:33:37

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