Calc BC AP Test Review

Volume of a Solid with Cross Section

• V=ʃ_a^b A(x)dx
• (cross section is perp. to height of solid)

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.)

Disc Method about the line h

• V= (pi) ʃ_a^b( f(x) - h )² dx,
• where | f(x)-h |=radius
• (if x-axis, h=0)

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

• V=(pi) ʃ_a^b [ (f(x))² - (g(x))² ]dx
• Outer radius - smaller radius

if g(x)< f(x) < h(x) and Lim_x->a[g(x)= h(x)]=L, then so does that same limit of f(x)

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

d/dx (a^u) = a^u ln(a) u'

d/dx log(u)= 1/(u lna) * u'

• 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

• 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

• 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

• 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

Average Value of a Function: 1/(b-a) ʃ_a^b f(x)dx

A=(s² 3^1/2) / 4

• Position Function = s(t)
• Instantaneous Velocity = s'(t) = v(t)
• Acceleration = s''(t) = v'(t) = a(t)
• Instantaneous Speed = |v(t)|

ʃ_a^b f(x)dx = F(b) - F(a)

F(x)= ʃ_a^b f(t)= f(x) , F'(x)=f(x) at every point of [a,b]

• ʃ_a^b [f(x)-g(x)]dx
• upper curve - lower curve

• Displacement from t₁ to t₂ = ʃ_t1^t2 v(t)
• Total distance traveled is the same thing but the absolute value of v(t)

• is a series of the form, 1+1/2^p=1/3^p=1/4^p+...+1/n^p+...= sum( 1/n^p, n, 1, infinity)
• Series converges when p>1
• Series diverges when 0<p<1

special p series, where p=1, so the series diverges

• sum(ar⁽ⁿ⁾,n,0,infinity).
• series converges when |r|<1
• sum of the series is a/(1-r)

if a_n=f(n) where f is continuous, positive, decreasing funciton on [c,infinity)