Calculus 2

Chapter 7
Logarithmic and Exponential Functions

Section 7.1

Definition {7.1} A function f with domain D and range R is a one-to-one function if whenever a does not equal b in D, then f (a) does not equal f (b) in R.

{7.5 (Guidelines for find f^-1^ in simple cases)}
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)]

Definition {7.2} 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 and every y in R: y = f (x) if and only if x = g (y)

Theorem {7.3} 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:
(i) g(f(x)) = x for every x in D
(ii) f(g(y)) = y for every y in R

{7.4 (Domains and Ranges of f and f^-1^) }
Domain of f^-1^=range of f
Range of f^-1^=domain of f

Theorem {7.6} 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)]

Theorem {7.7} If a differentiable function f has an inverse function g=f^-1^ and if f (g(c)) is not equal to 0, then g is differentiable at c and g' (c) = 1/ f '(g(c))

Corollary {7.8} If g is the inverse function of a differentiable function f and if f '(g(x)) is not equal to 0, then:
g'(x) = 1/f '(g(x))

Section 7.2

Definition {7.9} The natural logarithmic function , denoted by ln , is defined by: ln x = Integral from 1 to x (1/t) dt for every x > 0

Theorem {7.10} Dx lnx=1/x

Theorem {7.11} If u = g(x) and g is differentiable, then
(i) Dx ln u = 1/u if g(x)>0
(ii) Dx ln |u| = 1/u Dx u if g(x)is not equal to 0

{7.12} [Laws of natural logarithms] If p > 0 and q > o , then:
(i) ln pq = ln p + ln q
(ii) ln p / q = ln p - ln q
(iii) ln p ^ r = r ln p for every rational number r

{7.13} [Guidelines for logaritmic differentiation]
1. y = f (x) (given)
2. ln y = ln f (x) (take natural logarithms and simplify)
3. Dx [ ln y ] = Dx [ ln f (x) ] (differentiate implicitly)
4. 1 / y Dx y = Dx [ ln f (x) ] (by Theorem 7.11)
5. Dx y = f (x) Dx [ ln f (x) ] (multiply by y = f (x) )

Section 7.3

Theorem {7.14 } To every real number x there corresponds exactly one positive real number y such that ln y = x

Theorem {7.15} The natural exponential function , denoted by exp , is the inverse od the natural logarithmic function.

Theorem {7.20} If p and q are real numbers and r is a rational number, then:
(i) (e^p^) (e^q^)=e^p^ + q
(ii) (e^p^)/(e^q^)=e^p^ - q
(iii) (e^p^)^r^ = e^pr^

{7.16} [Definition of e] The letter e denotes the positive real number such that ln e = 1

{7.17} [Approximation to e] e approximately 2.71828

Theorem {7.22} If u = g(x) and g is differentiable, then:
Dx e^u^ = e^u^ Dx u

Theorem {7.19} (i) ln e^x^=x for every x
(ii) e^ln x^=x for every x>0

{7.18}[Definition of e^x^] If x is any real number, then
e^x^= y if and only if ln y=x

Theorem {7.21} Dx e^x^ = e^x^

Section 7.4

Theorem {7.23} If u = g (x) is not equal to 0 and g is differentiable, then:
Integral (1/u)du=ln|u| + C

Theorem {7.24} If u = g (x) and g is differentiable, then
integral (e ^ u) du = e ^ u + C

Theorem {7.25}
(i) Integral tan u du = - ln |cos u| + C
(ii) Integral cot u du = ln |sin u| + C
(iii) Integral sec u du = ln |sec u + tan u| + C
(iv) Integral csc u du = ln |csc u - cot u| + C

Section 7.5

{7.27}[Laws of exponents]

r

Let a > 0 and b > 0. If u and v are any real numbers, then (i) (a^u^)(a^v^) = a^u^ + v(ii) (a^u^)^v ^= a^uv^(iii) (ab)^u^ = (a^u^)(b^u^)(iv) a^u^ / a^v ^= a^u-v ^(v) (a/b)^u^ = a^u^ / b^u^

Theorem {7.28}

r

(i) Dx a^x^= a^x^ ln a(ii) Dx a^u^ = (a^u ln a ) Dx u

Theorem {7.29}

r

(i) Integral a^x^ dx = (1/ln a) a^x^ + C(ii) Integral a^u^ du = (1/ ln a) a^u^ + C

{7.26}[Definition of a^x^]

r

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

{7.30}[Definition of log_a_x]

r

y = loga x if and only if x = a^y^

Theorem {7.31}

r

(i) Dx loga x = Dx (ln x / ln a) = (1 / ln a)(1 / x)(ii) Dx loga |u| = Dx (ln |u| / ln a) = (1 / ln a)(1 / u) Dx u

Theorem {7.32}

r

(i) lim h=> 0 (1+h) ^1/h = e (ii) lim n=>infinity (1+1/n) ^n =e

Section 7.6

Theorem {7.33}

r

Let y be a differentiable function of t such that y > 0 for every t, and let y_0_ be the value of y at t = 0. If dy/dt = cy for soem constant c, theny = y0 e^ct^

Chapter 8
Inverse Trigonometric and Hyperbolic Functions

Section 8.1

{8.1} [Definition]

r

The Inverse sine function, denoted sin^-1^, is defined by y = sin^-1^ x if and only if x = sin y for -1 < or = x < or = 1 and -p/2 < or = y < or = p/2

{8.2} Properties of sin^-1^

r

(i) sin (sin^-1^ x) = sin (arcsin x) if -1 < or = x < or = 1(ii) sin^-1^ (sin x) = arcsin (sin x) if -p/2 < or = x < or = p/2

{8.3} [Definition]

r

The Inverse cosine function, denoted cos^-1^, is defined by y=cos^-1^ x if and only if x=cos y for -1 < or = x < or = 1 and 0 < or = y < or = p

{8.6} [Properties of tan ^-1^

r

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

{8.7} Defintion

r

The Inverse Secant Function, or arcsecant function, denoted by sec^-1, or arcsec, is defined byy = sec^-1^ x = arcsec x if and only if x = sec y for |x| > or = 1 and y in [0, p/2) or in [p, 3p/2)

{8.4} Properties of cos ^-1^

r

(i) cos (cos^-1^ x) = cos (arccos x) = x if -1 < or + x < or = 1(ii) cos^-1^ (cos x) = asrccos (cos x) = x if 0 < or = x < or = pie

{8.5} [Definition]

r

The Inverse Tagent Function denoted by tan^-1^, or arctan, is defined byy = tan^-1^ x = arctan x if and only if x = tan y for every x and -p/2 < y < p/2.

Section 8.2

Theorem {8.8}

r

(i) Dx sin^-1 u = [1 / rad (1 - u)^2] Dx u(ii) Dx cos^-1 u = [-1 / rad (1 - u)^2] Dx u(iii) Dx tan^-1 u = [1 / (1 + u)^2] Dx u(iv) Dx sec^-1 u = [1 / u rad (u^2 - 1)] Dx u

Theorem {8.9}

r

(i) Integral [1 / sq rt (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 sq rt(u^2 - a^2)] du = 1/a sec^-1 (u/a + C)

Section 8.3

{8.10} Definition

r

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^)/2for every real number x.

Theorem {8.11}

r

cosh^2^x-sinh^2^x=1

Theorem {8.13}

r

1-tanh^2 x = sech^2 xcoth^2 x - 1 = csch^2 x

Theorem {8.12}

r

(I) tanx = 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

Theorem {8.14}

a

Theorem {8.15}

a

Subtopic

Section 8.4

Theorem {8.16}

a

Theorem {8.17}

r

(I)dx sinh^-1 u=1/sqrt(u^2+1)dx u(II)dxcosh^-1 u= 1/sqrt(u^2-1) dx u u>1(III)dx tanh^-1u=1/(1-u^2) dx u [u]<1(IV)dx sexh^1u=-1/(usqrt(1-u^2) dx u 0<u<1

Theorem {8.18}

r

(I) int 1/sqrt(a^2+u^2)du=sinh^-1(u/a)+ca>0(II) int 1/sqrt(u^2-a^2)du=cosh^-1(u/a)+c0<a<u(III) int 1/(a^2-u^2)du=1/a tanh^-1 (u/a) +c[u]<a(IV) int 1/usqrt(a^2-u^2) du= -1/a sech^-1 ([u]/a) +c 0<[u]<a

Chapter 9
Techniques of Integration

Section 9.2

{9.2} Guidelines for evaluating

r

if m is odd in int. sin^m x cos^n x dx= int. sin^m-1 x cos^n x sinx dxwhere sin^m-1 is sin^2 x=1-cos^2 x, then sub u=cosxIf n isodd in int. sin^m x cos^n x dx= int. sin^m x cos^n-1 x cos xdxthen cos^2 x=1-sin^2 xthen sub u=sinx du=cosxdxif m and nare even use half angles for sin^2 x and cos^2 x

{9.3} Guidelines for evaluating

r

If m is odd in int. tan^m x sec^n xdx= int. tan^m-1 x sec^n-1 xsec x tan x dxexpress tan^m-1 as sec, make tan^s x =sec^2x-1u=sec x du=sec xtan x dxif n is an even integer int. tan^m xsec^n x dx= int tan^m x sec^n-2 x sec^2 x dxmake sec^2 x=tan^2 x +1 u=tan x du=sec^2 x dxif m is even and n odd thereis no form of solving

Section 9.3

{9.4} Trigonometric substitutions

r

sqrt a^2-x^2 ----------> x=a sin th sqrt a^2+x^2 ----------> x=a tan thsqrt x^2-a^2 -------------->x=a sec th

Section 9.4

{9.5} Guidelines for partial fraction decompositions

r

if degree of f x is lower than g x, use long divisionif a1/(x+1)(x-2)= a/(x+1) + b/(x-2)a1/(x^2+2)= ax+b/(x^2+2)a1/(x+7)^2= a/(x+7) + b/(x+7)^2

Section 9.1

{9.1} Integration by parts formula

r

If u = u(x), v = v(x), and the differentials du = u '(x) dx and dv = v'(x) dx, then integration by parts states that int u dv = uv - int v du.

Section 9.5 - Intregals involving quadratic expressions
REVIEW EXAMPLES 1-3

Section 9.6

Theorem {9.6}

r

If an integrand is a rational expression in sin x and cos x, the following substitutions will produce a rational expression in u:sin x = 2u/1 + U^2 , cos x = 1-u^2/1+u^2, dx = 2/1+u^2 du,where u = tan x/2

Section 9.7 - Tables of Intregals
REVIEW EXAMPLES 1-3

Chapter 10
Indeterminate Forms and Improper Integrals

Section 10.1

Cauchy's formula {10.1}

r

If f and g are continuous on [a, b] and differentiable on (a,b) and if g ' x is not 0 for every x in (a, b), then there is a number w in (a, b) such thatf(b) - f(a) = f ' (w)------------------------g(b) - g(a) = g ' (w)

L'hopital's Rule {10.2}

r

Suppose f and g are differentiable on an open interval (a, b) containing c, except possibly at c itself. If f(x)/g(x) has the indeterminate form 0/0 or inf/inf at x=c and if g'(x) is not 0 for x is not equal to c, then f(x) f ' (x) lim ---- = lim --------x->0 g(x) x->c g ' (x)provided either the ^^^^^^ (directly above the arrows) limit exists or the limit goes to infinity

Section 10.4

Defintion {10.7}

r

(I) If f is continuous on [a, b) and discontinuous at b, then b tint f(x) dx = lim int f(x) dx,a t->b- aprovided the limit exists.(II) If f is continuous on (a, b] and discontinuous at a, then b bint f(x)dx = lim int f(x)dx,a t->a+ tprovided the limit exists.

Definition {10.8}

r

If f has a discontinuity at a number c in the open interval (a, b) but is continuous elsewhere on [a, b], thenb c bint f(x)dx = int f(x)dx + int f(x)dxa a cprovided both of the improper intregals on the right converge. If both converge, then the value of the improper intregal b int a f(x) dx is the sum of the two values.

Section 10.3

Definition {10.5}

r

(I) If f is continuous on [a, inf), then inf tint f(x) dx = lim int f(x) dx, a t->inf aprovided the limit exists.(II) If f is continuous on (-inf, a], thena aint f(x)dx = lim int f(x) dx,-inf t->-inf tprovided the limit exists.

Definition {10.6}

r

Let f be continuous for every x. If a is any real number, then inf a infint f(x)dx = int f(x)dx + int f(x) d(x),-inf -inf aprovided both of the improper intregrals on the right converge.

Section 10.2

{10.3} Guidelines for Investigating for the form 0 times inf

r

1. Write f(x) g(x) asf(x)/1/g(x) or g(x)/1/f(x)2. Apply L'Hopital's Rule to the resulting indeterminate form 0/0 or inf/inf

{10.4} Guidelines for Investigating 0^0, 1^inf, inf ^0

r

1. let y = f(x)^g(x).2. Take natural logarithms in guideline 1:ln y = ln f(x)^g(x) = g(x) ln f(x)3. Investigate lim x-->c ln y = lim x-> c [g(x) ln f(x)] and conclude the following:a) If lim x->c ln y = L, then lim x->c y = e^Lb) If lim x->c ln y = inf, then lim ->c y = infc) If lim x->c lny = -inf, then lim x-c y = 0