Ashot H. Calculus II

A function f with domain D and range R is a one-to-one function if whenever a =! b in D, then f(a) =! f(b) in R.

Let f be one-to-one function with domain D and range R. A function g with domain R and range D is the inverse function of f, provided the following condition is true for every x in D every y in R: y=f(x) if and only if x=g(y).

Let f be a one-to-one function with domain D and range R. If g is a function with domain R and range D, then g is the inverse function of f if and only if both of the following conditions are true: 1) g(f(x)) = x for every x in D 2) f(g(y)) =y for every y in R.

domain of f^-1 = range of f ,, range of f^-1 = domain of f

If f is continuous and increasing on [a,b], then f has an inverse function f^-1 that is continuous and increasing on [f(a),f(b)].

If a differentiable function f has an inverse function g=f^-1 and if f'(g(c))=! 0, then g is differentiable at c and g'(c)=1/f'(g(c)).

If g is the inverse function of a differentiable function f and if f'(g(x)) =! 0, then g'(x) = 1/ f'(g(x)).

The natural logarithmic function, denoted by ln, is defined by ln x = (integral) from 1 to x of 1/t dt for every x > 0.

Dx ln x = 1/x

If u = g(x) and g is differentiable, then 1) Dx ln u = 1/u Dx u if g(x) > 0. 2) Dx ln |u| = 1/u Dx u if g(x) =! 0

If p>0 and q>0, then 1) ln (pq) = lnp + lnq 2) ln (p/q) = lnp - lnq 3) ln (p^r) = rlnp for every rational number r.

To every real number x there corresponds exactly one positive real number y such that ln y = x.

The natural exponential function, denoted exp, is the inverse of the natural logarithmic function.

The letter e denotes the positive real number such that ln(e) =1.

e = 2.71828

If x is any real number, then e^x = y if and only if ln y = x.

1) ln (e^x) = x for every x. 2) e^(lnx) = x for every x>0.

If p and q are real numbers and r is a rational number, then 1) e^(p) e^(q) = e^(p+q) 2) (e^p)/(e^q) = e^(p-q) 3) e^(p)r = e^pr.

Dx e^x = e^x

If u = g(x) and g is differentiable, then Dx e^u = e^u Dx u.

If u=g(x)=! 0 and g is differentiable, then (integral) of 1/u du = ln |u| + C.

If u = g(x) and g is differentiable, then (integral) e ^u = e^u +C.

1) (integral) tan(u) du = -ln|cos(u)| +C 2) (integral) cot(u) du = ln|sin(u)| + C 3) (integral) sec(u) du = ln|sec(u) + tan(u)| + C 4) (integral) csc(u) du = ln|csc(u) - cot(u)| + C.

a^x = e^(xlna) for every a>0 and ever real number x

Let a>0 and b>0. If u and v are any real numbers, then 1) a^(u) a^(v) = a^(u+v) 2) a^(u)v = a^(uv) 3) (ab)^u = a^(u) b^(u) 4) (a^(u))/(a^(v)) = a^(u-v) 5) (a/b)^u = (a^(u))/(b^(u)).

1) Dx a^x = (a^x)(lna) 2) Dx a^u = ((a^(u))(lna)) Dx u

1) (integral) of a^x dx = (1/(lna))(a^x) + C 2) (integral) of a^u du = (1/(lna))(a^u) + C

y = log(x) base a if and only if x = a^y

1) Dx log(x) base a = Dx((lnx)/(lna))= (1/(lna))*(1/x) 2) (integral) Dx log|u| base a = Dx ((ln|u|)/(lna))= (1/(lna))*(1/u) Dx u

1) lim(1+h)^(1/h) as h-> 0 = e 2) lim (1 + (1/n))^n as n-> (infinity) = e

Let y be a differentiable function of t such that y>0 for every t, and let y (sub 0) be the value of y at t=0. If dy/dt = cy for some constant c, then y = y(sub 0) e^(ct) .

The inverse sin function denoted sin^-1, is defined by y = sin^(-1)x if and only if x = siny for -1<= x <= 1 and -pi/2<= y <= pi/2.

i) sin(sin^-1x) = sin(arcsinx)=x if -1<= x <= 1 ii)sin^(-1)(sinx)= arcsin(sinx) = x if - pi/2 <= x <= pi/2.

The inverse cosine function, denoted cos^-1, is defined by y = cos^-1x if and only if x=cosy for -1 <= x <= 1 and 0 <= y <= pi.

i) cos(cos^-1x)=cos(arccosx)=x if -1 <= x <=1 ii) cos^-1(cosx)=arccos(cosx)=x if 0 <= y <= pi.

The inverse tangent function, or arctangent function, denoted tan^-1, or arctan, is defined by y=tan^-1x = arctanx if and only if x = tany for every x and -pi/2 <= y < = pi/2)

i)tan(tan^-1x)=tan(arctanx)=x for every x ii) tan^-1(tanx)= arctan(tanx)=x if -pi/2 <= x <= pi/2.

The inverse secant function, or arcsecant function, denoted by sec^-1 or arcsec, is defined by y = sec^-1x=arcsecx if and only if x =secy for |x|>= 1 and y in [0, pi/2) or in [pi, 3pi/2)

i) d/dx (sin^-1u) = (1/(sqrt(1-u^2))) du/dx ii) d/dx (cos^-1u) = -(1/(sqrt(1-u^2))) du/dx iii) d/dx (tan^-1u) = (1/(1+u^2) du/dx iv) d/dx (sec^-1u) = (1/(u(sqrt((u^2)-1)))) du/dx

i) (intagrel) (1/(sqrt((a^2)-u^2))) du = sin^-1(u/a) + C ii) (integral) (1/((a^2)+u^2)) du = (1/a)(tan^-1(u/a) + C iii) (integral) (1/(u(sqrt((u^2)-a^2) du = (1/a)(sec^-1(u/a) +C.

The hyperbolic sine function, denoted by sinh, and the hyperbolic cosine function, denoted by cosh, are defined by sinhx=((e^x)-(e^-x))/2 and coshx= ((e^x)+(e^-x))/2 for every real number x.

((coshx)^2) - ((sinhx)^2)=1

i) tanhx = sinhx/coshx = (e^x - e^-x )/ e^x +e^-x ii) cothx = coshx/sinhx = (e^x + e^-x)/ e^x - e^-x iii) sechx= 1/coshx = 2/(e^x + e^-x) iv) cschx = 1/sinhx = 2/(e^x - e^-x)

i) 1-(tanhx)^2 = (sechx)^2 ii) ((cothx)^2)-1 = (cschx)^2

i) d/dx sinhu = coshu du/dx ii) d/dx coshu = sinhu du/dx iii) d/dx tanhu = (sechu^2) du/dx iv) d/dx cothu = - (cschu^2) du/dx v) d/dx sechu = - (sechu)(tanhu) du/dx vi) d/dx cschu = - (cschu)(cothu) du/dx

i) (integral) sinhu du = coshu + C ii) (integral) coshu du = sinhu + C iii) (integral) sechu^2 du = tanhu + C iv) (integral) cschu^2 du = - cothu + C v) (integral) sechutanhu du = - sechu + C vi) (integral) cschucothu du = - cschu + C

i) sinh^-1 u = ln(x+(sqrt(x^2+1)) ii) cosh^-1 u = ln(x+(sqrt(x^2-1)) x>=1 iii) tanh^-1 u = (1/2)(ln((1+x)/(1-x))) |x|<1 iv) sech^-1 u = ln((1+(sqrt(1-x^2)))/x), 0<x<=1

i) d/dx sinh^-1 u = 1/(sqrt u^2+1)) du/dx ii) d/dx cosh^-1 u = 1/(sqrt u^2-1)) du/dx u>1 iii) d/dx tanh^-1 u = 1/(1-u^2)) du/dx |u| < 1 iv) d/dx sech^-1 u = - 1/((u)(sqrt 1-u^2))) du/dx 0 < u < 1

i) (integral) 1/(sqrt a^2+u^2)) du = sinh-1(u/a) + C, a>0 ii) (integral) 1/(sqrt a^2-u^2))du= cosh^-1(u/a + C, 0<a<u iii) (integral) 1/(a^2-u^2) du= (1/a)tanh^-1(u/a) + C, |u|<a iv)(integral)1/(u)(sqrt a^2-u^2)du=(-1/a)(sech^-1(|u|/a)+C

If u = f(x) and v = g(x) and if f' and g' are continuous, then (integral) u dv = uv - (integral) v du.

Miscellaneous substituions

If an integrand is a rational expession in sinx and cosx, the following substitusions will produce a rational expression in u : sinx= 2u/(1+u^2) cosx=(1-u^2)/(1+u^2), dx= (2/(1+u^2)) du where u = tan(x/2)

If f and g are continuous on [a,b] and differentiable on (a,b) and if g'(x)!=0 for every x in (a,b), then there is a number w in (a,b) such that ((f(b)-f(a))/(g(b)-g(a)))= (f'(w)/g'(w)) CAUCHY'S FORMULA

Suppose f and g are differentialbe on an open interval (a,b) containing c, except possible at c itself. If f(x)/g(x) has the indeterminate for 0/0 or (infinity/infinity) at x = c and if g'(x)!=0 for x !=c, then limit (f(x)/g(x)) as x->c = limit f'(x)/g'(x) as x->c provided either limit f'(x)/g'(x) as x->c exists or limit f'(x)/g'(x) as x->c = (infinity).

i) If f is continous on [a, (infinity)), then (integral) f(x) dx from a - (infinity) = limit (integral) f(x) dx from a - t as t-> (infinity) provided the limits exist. ii) If f is continous on (-infinity,a], then (integral) f(x) dx from -infinity to a = limit (integral) f(x) dx from t - a as t -> (-infinity) provided the limits exists.

Let f be continuous for every x. If a is any real number, then (integral) f(x) dx from (-infinity to infinity) = (integral) f(x) dx from -infinity to a + (integral) f(x) dx from a to infinity, provided both of the improper integrals on the right converge.

i) If f is continuous on [a,b) and discontinuous at b, then (integral) f(x) dx from a - b = limit (integral) f(x) dx from a - t as t -> b^- provided the limits exists. ii) If f is continuous on (a,b] and discontinous at a, then (integral) f(x) dx from a - b = limit (integral) f(x) dx from t - b as t ->a^+ provided the limits exists.

If f has a discontinuity at a number c in the open interval (a,b) but is continuous elsewhere on [a,b], then (integral) f(x) dx from a - b = (integral) f(x) dx from a-c + (integral) f(x) dx from c-b, provided both of the improper integrals on the right converge. If both converge, then the value of the improper integral (integral) f(x) dx from a -b is the sum of the two values.

A sequence is a function f whose domain is the set of positive integers.

A sequnce {a_n) has the limit L, or converges to L, dentoed by either limit (a_n) = L as n-> (infinity) or a_n -> L as n -> (infinity), if for every (epsilon) > 0 there exites a positive number N such that |(a_n)-L| < (epsilon) whenver n>N. If such a number L does not exist, the sequence has no limit, or diverges.

The notation limit (a_n) = (infinity) as n -> (infinity) means that for every positive real number P there exists a number N such that (a_n) > P whenver n > N.

Let {a_fn} be a sequence , let f(n) = (a_n), and suppose that f(x) exists for every real number x >= 1. i) If limit f(x) = L as x -> (infinity), then limit f(n) = L as n -> (infinity) ii) If limit f(x) = (+ or - infinity) as n -> (infinity) , then limit f(n) = (+ or - infinity) as n -> (infinity).

i) limit (r^n) = 0 as n -> (infinity) if |r|<1. ii) limit (r^n) = (infintiy as n -> (infinity) if |r| > 1.

If {a_n},{b_n}, and {c_n} are sequences and a_n <= b_n <= c_n for every n and if limit a_n = L = limit c_n as n -> (infinity) then limit b_n = L as n -> (infinity).

Let {a_n} be a sequence. If limit |a_n| = 0, then limit a_n = 0 as n -> (infinity).

A bounded monotonic sequence has a limit

If a nonempty set S of real numbers has an upper bound, then S has a least upper bound.

An infinite series (or simply a series) is an expression of the form a1 + a2 + ..... + a_n or in summation notation, (sigma) a_n . Each number a_k is a term of the series, and a_n is the nth term.

i) The kth partial sum S_k of the series (sigma) a_n is S_k = a1 + a2 + ... + a_k. ii) The sequance of partial sums of the series (sigma) a_n is S_1, S_2, S_3,........S_n,.......

A series (sigma) a_n is convergent if its sequence of partial sums converges that is, if limit S_n = S for some real number S. THe limit S is the sum of the series (sigma) a_n and we write S = a_1 + a_2 +.....+a_n+....... The series (sigma) a_n is divergent if S_n diverges. A divergent series has no sum.

The harmonic series is the divergent series 1 + 1/2 + 1/3 +.......+ 1/n +.....

Let a=! 0, The geometric series a + ar+ ar^2 + ..... + ar^n-1 + .... i) converges and has the sum S= a/(1-r) if |r| <1 ii) diverges if |r| >=1.

If a series (sigma) a_n is convergent, then limit a_n = 0.

i) If limit a_n =! 0, then the series (sigma) a_n is divergent. ii) if limit a_n = 0, then further inverstigation is neccessary to determine whether the series (sigma) a _n is convergent or divergent.

If (sigma) a_n and (sigma) b_n are series such that a_j = b_j for every f> k, where k is a positive integer, then both series converge or both series diverge.

For any positive integer k, the series (sigma) a_n = a_1 +a_2 + .... and (sigma) (a_n) = a_k+1 + a_k+2 + ...... either both converge or both diverge.

If (sigma) a_n and b_n are convergent series with sums A and B, respectively, then i) (sigma) (a_n + b_n) converges and has sum A+B ii)(sigma) (ca_n) converges and has sum cA for every real number c iii) (sigma) (a_n-b_n) converges and has sum A - B

If (sigma) a_n is a convergent series and (sigma) b_n is divergent, then (sigma) (a_n + b_n) is divergent.

If (sigma) a_n is a positive term series and if there exits a number M such that S_n = a_1 + a_2 + ... .+ a_n < M for every n, then the series converges and has sum S <= M. if no such M exits, the series diverges.

If (sigma) a_n is a series, let f(n) = a_n and let f be the function obtained by replacing n with x. If f is a positive valued, continuous and decreasing for every real number x>=1, then the series (sigma) a_n i) conveges if (integral) f(x) dx converges from 1 - infinity ii) divergees if (integral) (fx) dx diverges from 1 - infinity

A p series or a hyperharmonic series, is a series of the form (sigma) 1/n^p (n=1 to infinity) = 1 + 1/2^p + 1/3^p + ... + 1/n^p + ..., where p is a positive real number

The p series (sigma) 1/n^p i) converges if p >1 ii) diverges if p<= 1

Let (sigma) a_n and b_n be positive term series. i) if (sigma) b_n converges and a<b for every positive integer n, then (sigma) a_n converges. ii) If (sigma) b_n diverges and a>b for every positive number n, then (sigma) a_n diverges.

Let (sigma) a_n and b_n be positive term series. If limit a_n/b_n = c > 0 as n-> infinity, then either both series converge or both diverge.

Let (sigma) a_n be a positive term series, and suppose lim a_n+1/a_n = L as n -> infinity. i) If L<1, the series is convergent. ii) If L>1 or limit (a_n+1/a_n) = infinity, series is divergent. iii) If L = 1, apply a different test; the series may be convergent or divergent

Let (sigma) a_n be a positive term series and suppose limit (a_n)^1/n = L for n -> infnity. i) If L < 1, series is convergent. ii) If L > 1 or limit (a_n)^1/n = infnity, series is divergent. iii) If L = 1, apply a different test; the series may be convergent or divergent

The alternating series (sigma) (-1)^n-1 a_n = a1-a2+a3-a4 + .........+ (-1)^n-1 a_n + ......... is convergent if the following two conditions are satisfied i) a_k >= a_k+1 > 0 for every k ii) limit a_k = 0 as n -> infnity

Let (sigma) (-1)^n-1 a_n be an alternating series that satisfies conditions (i) and (ii) of the alternating series test. If S is the sum of the series and S_n is a partial sum, then |S-S_n| <= a_n+1; that is, the error involved in approximating S by S_n is less than or equal to a_n+1.

A series (sigma) a_n is absolutely convergent if the series (sigma) |a_n| = |a1| + |a2| + ...... + |a_n| + ....... is convergent.

A series (sigma) a_n is conditionally convergent if (sigma) a_n is convergent and (sigma) |a_n| is divergent.

If a series (sigma) a_n is absolutely convergent, then (sigma) a_n is convergent.

Let (sigma) a_n be a series of nonzero terms, and suppose limit |(a_n+1)/(a_n)| = L as n -> infinity i) If L < 1, the series is absolutely convergent. ii) If L > 1, or limit = infinity the series is divergent. iii) If L = 1, apply a different test; the series may be absolutely convergent, conditionally convergent , or divergent.

Let x be a variable. A power series in x is a series of the form (sigma) n = 0 to infnity (a_n)x^n = a_0 + a_1x + a_2x^2 + ...... + a_n x^n + ........ where each a_k is a real number.

i) If a power series (sigma) a_n x^n converges for a nonzero number c, then it is absolutely convergent whenver |x| < |c|. ii) If a power series (sigma) a_n x^n diverges for a nonzero number d, then it diverges whenever |x| > |d|.

If (sigma) a_n x^n is a power series, then only one of the following is true: i) the series converges only if x = 0 ii) the series is absolutely convergent for every x. iii) there is a number r > 0 such that the series is absolutely convergent if x is in the open interval (-r,r) and divergent if x < -r or x > r.

Let c be a real number and x a variable. A power series in x-c is a series of the form (sigma) n=0 to infnity a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + ...... + a_n(x-c)^n + .. where each a _k is a real number.

Suppose a power series (sigma) a_n x^n has a readius of convergence r > 0 and let f be defined by f(x) = (sigma) a_n x^n = a_0 + a_1x + a_2x^2 + a_3x^3 +......+ a_nx^n +.. for every x in the interval of convergence. If -r<x<r, then i)f'(x)=a_1+2a_2x+3a_3x^2+....+na_nx^n-1+...=(sigma)na_nx^n-1 n = 1 to infinity ii) (integral) 0-infinity f(t) dt = a_0x + a_1(x^2)/2 + a_2(x^3)/3 + .... +a_n(x^n+1)/n+1 + .... = (sigma) n = 0 to infinity (a_n/(n+1)) x^(n+1)

If x is any real number, e^x= 1+x+(x^2)/2! + (x^3)/3! + .... + x^n/n!+......

If a function has a power series representation f(x) = (sigma) n=0 to infnity (a_n)x^n with a radius of convergence r > 0, then f^(k)(0) exists for every positive integer k and a_n=f(n)(0)/n!. Thus, f(x)=f(0)+f'(0)x + f''(0)x^2/2!+....+f^(n)(0)x^n/n!+.......

If a function f has a power series representation f(x)=(sigma) n=0 to infninty a_n (x-c)^n with radius of convergence r > 0, then f^(k)(c) exists for every positive integer k and a_n=f(n)(c)/n!. Thus, f(x)=f(c)+f'(c)(x-c) + f''(0)(x-c)^2/2!+....+f^(n)(c)(x-c)^n/n!+.......

Let c be a real number and let f be a function that has n derivatives at c: f'(c), f"'(c). The nth-degree Talor polynomial p_x(x) of f at c is P_n(x)=f(c) +f'(c)(x-c)+f''(c)(x-c)^2+...+f^(n)(c)(x-c)^n/(n!).

Let f have n+1 derivatives throughout an interval containing c. If x is any number in the interval that is different from c, then there is a number z between c and x such that f(x) = p_n(x)+R_n(x), where R_n(x)=(f^(n+1)(z))(x-c)^n+1/(n+1)!

Let f have derivatives of all odrderes throughout an interval containing c, and let R_n(x) be the Talor remainder of f at c. If limit R_n(x)=0 as n -> infinty for every x in the interval, then f(x) is represented by the Taylor series for f(x) at c.

If x is any real number. limit |x|^n/n! = 0 as n -> infinity

Let f have n+1 derivatives throughout an interval containing c. If x is any number in the interval and x=! c, then the error in the approximating f(x) by the nth-degree Taylor polynomial of f at c, P_n(x)=f(c)+f'(c)(x-c)+f''(c)(x-c)^2/2!+.....+f^(n)(c)(x-c)^n/n!, is |R_n(x)|, where R_n(x) = f^(n+1)(z)(x-c)^n+1/(n+1)! and z is the number between c and x given by (11.45).

If |x|<1, then for every real number k, (1+x)^k=1+kx+k(k-1)x^2/2! +.....+( k(k-1)...(k-n+1)(x^n) )/n! +...

If (x,y) are the coordinates of a point P in an xy plance if (x',y') are the coordinates of P in an x'y' plane with origin at the point (h,k) of the xy plane, then
i) x = x' + h y = y'+k
ii) x'= x-h y' = y-k

The graph of the equation
(x^2)/(a^2) + (y^2)/(b^2) = 1
for a^2>b^2 is an ellipse with vertices (+- a,0). The endopoints of the minor axis are (0,+-b). The foci are (+-c,0), where c^2=a^2-b^2.

The graph of the equation
(x^2)/(b^2) + (y^2)/(a^2) = 1
for a^2>b^2 is an ellipse with vertices (0,+- a). The endopoints of the minor axis are (+-b,0). The foci are (0,+-c), where c^2=a^2-b^2.

The eccentricity e of an ellipse is
e=c/a=(sqrt(a^2-b^2))/a

The graph of the equation
(x^2)/(a^2) - (y^2)/(b^2) = 1
is a hyperbola with vertices (+- a,0). The foci are (+-c,0), where c^2=a^2 + b^2.

The graph of the equation
(y^2)/(a^2) - (x^2)/(b^2) = 1
is a hyperbola with vertices (0,+- a). The foci are (0,+-c), where c^2=a^2 + b^2.

If the x and y axes are rotated about the origin O, through an acute angle (phi), then the coordinates (x,y) and (x',y') of a point P in the xy and x'y' planes are related as follows
i) x=x'cos(phi)-y'sin(phi), y=x'sin(phi)+y'cos(phi)
ii) x'=xcos(phi)+ysin(phi), y'=-xsin(phi)+ycos(phi)

To eliminate the xy term from the equation
Ax^2+Bxy+Cy^2+Dx+Ey+F=0, where B!=0, choose an angle (phi) such that cos(2(phi))= (A-C)/B with 0<(2phi)<180.. and use the rotation of axes formulas.

The graph of the equation
Ax^2+Bxy+Cy^2+Dx+Ey+F=0
is either a conic or a degenerate conic. If the graph is a conic, then it is
i) a parabola if B^2-4AC=0
ii) an ellipse if B^2-4AC<0
iii) a hyperbola if B^2-4AC>0

A plane curve is a set C of ordered pairs (f(t), g(t)), where f and g are continous functions on an interval I.

Let C be the curve consisting of all ordered pairs (f(t),g(t)). where f and g are continuous on an interval I. The equations
x=f(t), y=g(t)
for t in I, are parametric equations for C with parameter t.

If a smooth curve C is given parametrically by x = f(t), y = g(t), then the slope dy/dx of the tangent line to C at P(x,y) is dy/dx=(dy/dt)/(dx/dt). provided dx/dt !=0.

If a smooth curve C is given parametrically by x = f(t), y = g(t); a <t < b, and if C does not intersect itself, except possibly for t=a and t = b, then the length L of C is
L= (integral) (sqrt((f'(t))^2+(g'(t))^2) dt from (a-b) = (integral) (sqrt((dx/dt)^2+(dy/dt)^2) dt from (a-b)

dx = sqrt(dx^2+dy^2)= sqrt((dx/dt)^2+(dy/dt)^2) dt

Let a smooth curve C be given by x = f(t), y = g(t); a < t < b, and suppose C does not intersect itself, except possibly at the piont corresponding to t=a and t=b. If g(t) >0 throughout [a,b], then the area S of the surface of revolution obtained by revolving C about the x-axis is
S= (integral) 2(pi)ydx from (t=a - t=b) = (integral) 2(pi)g(t)(sqrt((dx/dt)^2+(dy/dt)^2)) dt from (a-b)

The rectangular coordinates (x,y) and polar coordinates (r,0) of a point P are related as follows:
i) x = rcos(theta), y = rsin(theta)
ii) r^2=x^2+y^2, tan(theta)=y/x if x !=0

i) The graph of r = f(theta) is symmetric with respect to the polar axis if substitutoin of - (theta) for (theta) leads to an equivalent equation.
ii) The graph of r = f(theta) is symmetric with respect to the vertical line (theta)= pi/2 if substitution of either (a) pi - theta for theta or (b) -r for r and - theta for theta leads to an equivalent equation.
iii) The graph of r = f(theta) is symmetric with respect to the pole if substitution of either (a) -r for r or (b) pi + theta for theta leads to an equivalent equation.

The slope m of the tangent line to the graph of r = f(theta) at the point P(r,theta) is
m = ((dr/d(theta))sin(theta) + rcos(theta))/((dr/d(theta))cos(theta) - rsin(theta)

If f is continuous and f(theta) > 0 on [(alpha),(beta)], where 0<alpha<beta<2pi, then the area A of the region bounded by the graphs of r = f(theta), theta = alpha, and theta = beta is
A= (integral) 1/2((f(theta))^2) d(theta) from alpha to beta = (integral) 1/2(r^2) d(theta) from alpha to beta.

ds= sqrt(r^2+(dr/d(theta))^2) d(theta)

About the polar axis: S= (integral) 2(pi)yds from (alpha - beta)= (integral) 2(pi)rsin(theta) ds from (alpha-beta)
About the line (theta)=pi/2: S = (integral) 2(pi)xdsfrom (alpha - beta)= (integral) 2(pi)rcos(theta) ds from (alpha-beta)

Let F be a fixed point and l a fixed line in a plane. The set of all points P in the plane, such that the ratio d(P,F)/d(P,Q) is a positive constant e with d(P,Q) the distance from P to l, is a conic section. The conic is a parabola if e=1, an ellipse if 0<e<1, and a hyperbola if e>1.

A polar equation that has one of the four forms
r = de/(1+-ecos(theta), r = de/(1+-esin(theta) is a conic section. Th conic is a parabola if e=1, an ellipse if 0<e<1, or a hyperbola if e>1.

A first order linear differential equation is an equation of the form
y' + P(x)y= Q(x), where P and Q are continuous functions.

The first order linear differential equation y' + P(x)y=Q(x) may be transformed into a separable differential equation by multiplying both sides by the integrating factor e^(integral)P(x)dx.