Download Calc2_RV1

MAC 2312 – Calculus II
REVIEW FOR TEST #1
FLORIDA INTERNATIONAL UNIV.
SPRING 2012
REMEMBER TO BRING AN 8x11 BLUE EXAM BOOKLET FOR THE TEST
Relevant sections of the textbook: [Review Sec. 5.1-5.3], Ch.5: Sec 5.4-5.10 & Ch.6: Sec. 6.1-6.6
MAIN PROBLEM SOLVING TECHNIQUES:
1.
How to find the value of various kinds of finite sums involving n.
2.
How to find the area under a simple curve by using circumscribed or inscribed rectangles &
taking limits as n→ ∞; and how to check your answers by using integration.
3.
How to evaluate integrals by using anti-derivatives & the Fundamental theorem of calculus.
4.
How to find the derivatives of integrals with limits that depend upon x.
5.
How to find the average value, fave , of a function f over [a,b] and how to find a value xave
such that fave = f(xave).
6.
How to the net displacement & the total distance travelled by a particle with velocity v(t).
7.
How to change one definite integral into another definite integral by using a substitution.
8.
How to prove the properties of ln(x) by using the definition of ln(x) and how to prove the
properties of ex by using the fact that ex is the inverse function of ln(x).
9.
How to find the area between two curves by using vertical strips or horizontal strips.
10.
How to find the volume of a solid of revolution by using thin washers or cylindrical shells.
11.
How to find the length of a plane curve or the surface area of a surface of revolution.
12.
How to find the work done in stretching springs or removing fluids from a container.
KEY CONCEPTS AND MAIN DEFINITIONS:
Finite sums, index of summation, Partitions of [a,b], Selection from a partition, Riemann
sums, the definite integral of over [a,b], anti-derivatives of f over [a,b], Fundamental theorem of
calculus parts (a) & (b), the average value of f over [a,b]; velocity, speed & acceleration of a
particle, the net displacement & the total distance travelled by a particle in a straight line, the natural
logarithmic function ln(x), the natural exponential functions exp(x) = ex, smooth curves, area,
volume, arc-length and surface area; work done by a varying force or by a force exerted for varying
distances.
MAIN FORMULAS AND THEOREMS:
1.
1 + 2 + 3 + . . . + n = n(n+1)/2,
12 + 22 + 32 + . . . + n2 = n(n+1)(2n+1)/6,
3
3
3
3
2
1 + 2 + 3 + . . . + n = {n(n+1)/2} ,
1 + r + r2 + r3 + . . . + rn = (rn+1 - 1) / (r-1).
2.
∫ab f(x).dx = lim max ∆xi →0 ∑ i=1n f(xi*) . ∆xi , where each xi* is in [xi-1, xi] .
3.
(d/dx) {∫ax f(t) dt} = f(x),
∫ab f(x) dx = [(d/dx)-1 f(x)]ab = [ ∫ f(x).dx]ab
4.
fave = ∫ab f(x) dx /{b-a},
∫ab f{g(x)}. {g’(x)}. dx = ∫g(a)g(b) f(u).du
b
5.
Net displacement = ∫a v(t).dt,
Total distance travelled = ∫ab |v(t)|.dt,
6.
ln(1/a) = - ln(a), ln(a.b) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), ln(ar) = r.ln(a).
7.
e-a = 1/ea, ea+b = ea.eb, ea-b = ea/eb, ea.r = (ea)r, ln (ea)=a, eln(a) = a, ex =y iff x=ln(y).
8.
A1 = ∫ab {y2(x) - y1(x)}.dx
A2 = ∫cd {x2(y) - x1(y)}.dy
b
2
2
9.
Vx = ∫a {π.(y2) - π.(y1) }. dx
Vy = ∫ab 2π. x .{y2(x) - y1(x)}. dx
10.
11.
12.
L1 = ∫ab √{1 + (dy/dx)2}. dx
L2 = ∫cd √{1 + (dx/dy)2}. dy
Sx = ∫ab 2π. y .√{1 + (dy/dx)2}.dx
Sy = ∫ab 2π. x .√{1 + (dy/dx)2}.dx
Wspring = ∫ab force(x).dx , Wfluid = ∫ab (distance fluid moves).(density of fluid).g.dV ,
where g = 10 ms-2 is the acceleration due to gravity and dV = volume of slice moved.