cosh 2 x sinh 2 x = 1 sin 2 x = 1 2 cos 2 x = 1 2 dx = dt r 2 = x 2 + y 2 L =

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1 Integrals Volume: Suppose A(x) is the cross-sectional area of the solid S perpendicular to the x-axis, then the volume of S is given by V = b a A(x) dx Work: Suppose f(x) is a force function. The work in moving an object from a to b is given by: W = b dx = ln x + C x tan x dx = ln sec x + C a f(x) dx sec x dx = ln sec x + tan x + C a x dx = ax ln a + C for a Integration by Parts: u dv = uv Arc Length Formula: L = b d d (sinh x) = cosh x dx a v du + [f (x)] 2 dx Derivatives Inverse Trigonometric Functions: d dx (sin x) = x 2 d dx (cos x) = x 2 d dx (tan x) = (cosh x) = sinh x dx d dx (csc x) = d dx (sec x) = d + x 2 dx (cot x) = x x 2 x x 2 + x 2 If f is a one-to-one differentiable function with inverse function f and f (f (a)) 0, then the inverse function is differentiable at a and (f ) (a) = f (f (a)) Hyperbolic and Trig Identities Hyperbolic Functions sinh(x) = ex e x 2 cosh(x) = ex + e x 2 tanh(x) = sinh x cosh x cosh 2 x sinh 2 x = sin 2 x = 2 ( cos 2x) cos 2 x = 2 ( + cos 2x) sin(2x) = 2 sin x cos x csch(x) = sinh x sech(x) = cosh x coth(x) = cosh x sinh x sin A cos B = 2 [sin(a B) + sin(a + B)] sin A sin B = 2 [cos(a B) cos(a + B)] cos A cos B = 2 [cos(a B) + cos(a + B)] dy dy dx = dt dx dt if Arc Length: L = x = r cos θ r 2 = x 2 + y 2 Area: A = L = b a b a r 2 + Parametric dx dt 0 β α (dx ) 2 + dt Polar y = r sin θ tan θ = y x 2 r(θ)2 dθ ( ) dr 2 dθ dθ ( ) dy 2 dt dt

2 Series nth term test for divergence: If lim a n does n not exist or if lim a n 0, then the series a n n is divergent. The p-series: and divergent if p. Geometric: If r < then is convergent if p > np ar n = n=0 a r The Integral Test: Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f(n). Then (i) If (ii) If then then f(x) dx is convergent, a n is convergent. f(x) dx is divergent, a n is divergent. The Comparison Test: Suppose that a n and b n are series with positive terms. (i) If b n is convergent and a n b n for all n, then a n is also convergent. (ii) If b n is divergent and a n b n for all n, then a n is also divergent. The Limit Comparison Test: Suppose that an and b n are series with positive terms. If a n lim = c n b n where c is a finite number and c > 0, then either both series converge or both diverge. Alternating Series Test: If the alternating series ( ) n b n satisfies (i) 0 < b n+ b n (ii) lim b n = 0 n for all n then the series is convergent. The Ratio Test (i) If lim a n+ n a n = L <, then the series a n is absolutely convergent. (ii) If lim a n+ n a n = L > or lim a n+ n a n =, then the series a n is divergent. (iii) If lim a n+ n a n =, the Ratio Test is inconclusive. f (n) (0) Maclaurin Series: f(x) = n! n=0 x n Taylor s Inequality If f (n+) (x) M for x a d, then the remainder R n (x) of the Taylor series satisfies the inequality R n (x) M (n + )! x a n+ for x a d Some Power Series e x x n = n! n=0 sin x = cos x = ( ) n x2n+ (2n + )! n=0 n=0 ln( + x) = ( ) n x2n (2n)! ( ) n xn n R = R = R = R = x = x n R = n=0

3 Practice Final Version A, Sec6 Q[Sec5.2, rotating solid] (a)[horizontal axis ] Sketch the region R bounded by y = e x, y = 0, x = 0, x = 2. Set up an integral for the volume of the solid rotating R about the x axis. Do not evaluate the integral. (b)[vertial axis ] Sketch the region R bounded by y = e x, x = 0, y = 2. Set up an integral for the volume of the solid rotating R about the y axis. Do not evaluate the integral. Q2[Sec5.4, Water-Pumping] A conical water tank with a top diameter of 8 feet and height of 0 feet is standing at ground level as shown in the sketch below. Water weighing 60 pounds per cubic foot is pumped from the tank to an outlet 3 feet above the top of the tank. If the tank is full, how many foot-pounds of work are required to pump all of the water from the tank?

4 Q3[Sec6., derivative formula for inverse functions] Let f(x) = x ln x + x 2, x > 0. Find (f ) (0). Hint: f() = 0. Q4[Sec , exp/log functions] Find the derivative of the following functions. (a)[exp-differential rule ] f(x) = (sec x) ln(x2 +) (b)[log-property ] x f(x) = ln ( tan x ) Q5[Sec6.5/9.3, initial value problem] Consider the following differential equation Find the solution of y = y(x) sec x dy dx y = 0, y(0) = 4

5 Q6[Sec6.8/0., L Hospital s Rule ] Evaluate the following limits. (a)[ 0 -type ] n n2 + lim n (b)[ 0-type ] lim n ln ( cos ) n n (c)[ / -type or leading term rule ] n lim 2 n 2n4 + 2

6 Q6[Sec7.-7.2, integration by parts and trig-integral] Evaluate the following integrals. (a)[sec7.,ibp for polynomial sin / cos /exp-type] (x + ) sin x dx (b)[sec7.,ibp for ln / tan / sin -type] (2x + ) ln x dx (c)[sec7.2, Odd/Even rule for sin cos-type] π/6 (2 + cos θ) 2 dθ 0

7 Q7[Sec , trig-sub and partial fraction decomposition] Evaluate the following integrals. (a)[u-sub VS trig-sub] x 3 x 2 dx (b)[sec7.3, sin formula] 00 dx 9 25x 2 (c)[sec7.4, P.F.D. linear product type] 2 t 2 dt

8 Q8[Sec7.8, improper integral] Determine whether each of the improper integral is convergent or divergent. Evaluate the improper integral if it is convergent. (a)[critical at ] 4x x dx 2 (b)[critical at finite ] /2 dx 0 ( 2x) /3 Q9[Sec8., arc-length formula] Find the arc-length of the curve y = 2 3 (x + )3/2 from x = to x = 2.

9 Q0[Sec.2 ] Determine whether each of the series is convergent or divergent. Please show your work and name any test(s) that are used. (a)[sec.2,n-th term test for DIV] cos ( 3n ) 2 (b)[sec.4,(limit) Comparison Test] sin ( 3n ) 2 (c)[sec.4,(limit) Comparison Test] n 2n4 + 2 (d)[sec.6, Ration Test] n 2 n

10 Q[Sec.6,.8, ratio test for the radius of convergence of power series] Consider the following power series. Find its center and radius of convergence. 3 n (x + 5) n n + Q2[Sec.9/.0, power series representation and its integral] Let F (x) = x t dt. Find the first four non-zero terms of the Maclaurin series for F (x) and F ( x).

11 Q2, Taylor series Let f(x) = cos(x). Consider its Taylor series at x = π/4. (a)[sec.0, n-th degree Taylor polynomial, derivative table ] Find T 3 (x), the 3rd degree Taylor polynomial of f(x) centered at π/4. (b)[sec., Taylor s Inequality ] Use Taylor s Inequality to estimate the maximum possible error in approximating f(x) by T 3 (x) for x [0, π/2].

12 Q3[Sec0.,0.2, derivative formula for parametric equations] Find the tangent line to the parametric curve x(t) = ln ( t + ), y(t) = t + 3 at t = t Q4[Sec0.3,0.4, polar curve] Consider the following polar curves given by r = 2( + sin θ), and r 2 = 2. (a)[circle and Cardioid in polar coordinates ] Sketch both curves r and r 2. (b)[area in polar coordinates ] Set up the integral for the area of the region bounded by r, θ = 0 and θ = π. (Do not evaluate.) 4 (c)[region bounded by two polar curves ] Find the area of the region inside r and outside r