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P E A R S ON ’ S

Calculus Review, Single Variable

Differentiation USEFUL DERIVATIVES

DERIVATIVE The derivative of the functionf (x) with respect to the variable x is the function f ¿ whose value at x is

Limits LIMIT LAWS

The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number.

If L, M, c, and k are real numbers and lim f (x) = L and

x:c

1.

lim g(x) = M, then

7.

If P(x) = an x n + an- 1x n- 1 + % + a0 is a polynomial then lim P(x) = P(c) = ancn + an- 1cn- 1 + % + a0 .

8.

If P(x) and Q (x) are polynomials and Q (c) Z 0 , then the rational P(c) P(x) P(x) = function has lim . Q(x) Q(c) x : c Q(x)

x:c

x:c

Sum Rule: lim ( f (x) + g(x)) = L + M x:c

The limit of the sum of two functions is the sum of their limits. 2.

Difference Rule: lim ( f (x) - g(x)) = L - M x:c

USEFUL LIMITS

The limit of the difference of two functions is the difference of their limits. 3.

1.

Product Rule: lim ( f (x) – g (x)) = L– M x:c

2.

The limit of a product of two function is the product of their limits. 4.

6.

z:x

q E -q(n(neven) odd) ,

lim

x : ;q

FINDING THE TANGENT TO THE CURVE y = f (x) AT (x0, y0)

1 = 0 xn

lim

Calculate f (x0) and f (x0 + h) .

2.

f (x0 + h) - f (x0) Calculate the slope f ¿(x0) = m = lim . h h:0 If the limit exists, the tangent line isy = y0 + m(x - x0) .

1.

Constant Rule: If f (x) = c (c constant), thenf ¿(x) = 0 .

2.

Power Rule: If r is a real number,

3.

d Constant Multiple Rule: (c – f (x)) = c– f ¿(x) dx

4.

Power Rule: If r and s are integers with no common factor r, s Z 0, then

5.

lim ( f (x))

x:c

provided that L L 7 0.)

r/s

= L

an x n + an- 1x n- 1 + % + a0 an = – lim x n- m bm x m + bm - 1x m - 1 + % + b0 bm x : ; q

4.

Sum Rule:

5.

Product Rule:

(provided an, bm Z 0)

The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.

r/s

For integers n, m > 0 , x:; q

f (x) L , M Z 0 = M x : c g(x )

Quotient Rule: lim

r/s

lim;

x:c

is a real number. (If s is even, we assume that

1 (n even) = E +q ;q (n odd) (x - c)n

6.

sin x sin kx lim x = 1, lim x = k (k constant), x:0

x:0

lim

x:0

2.

d (cos x) = -sin x dx

3.

d (tan x) = sec2 x dx

4.

d (cot x) = - csc2 x dx

5.

d (sec x) = sec x tan x dx

6.

d (csc x) = - csc x cot x dx

7.

d x (e ) = e x dx

9.

1.

x:c

3.

f (z) - f (x) . z - x

d (sin x) = cos x dx

11.

7.

cos x - 1 = 0 x

d xr = rx r - 1 dx

d [ f (x) ; g(x)] = f ¿(x) ; g¿(x) dx

d [ f (x)g(x)] = f (x)g¿(x) + f ¿(x)g(x) dx g(x) f ¿(x) - f (x)g¿(x) d f (x) c d = Quotient Rule: dx g(x) [g(x)]2 Chain Rule:

dy du d [ f (g(x))] = f ¿(g(x)) – g¿(x) = – du dx dx

if y = f (u) and u = g(x)

8.

d x (a ) = (ln a)a x dx

1 d (ln x) = x dx

10.

1 d (loga x) = x ln a dx

1 d (sin-1 x) = dx 11 - x 2

12.

-1 d (cos-1 x) = dx 11 - x 2

1 d (tan-1 x) = dx 1 + x2

14.

-1 d (cot -1 x) = dx 1 + x2

15.

1 d (sec-1 x) = dx ƒ x ƒ 1x 2 - 1

16.

-1 d (csc - 1 x) = dx ƒ x ƒ 1x 2 - 1

d (sinh x) = cosh x dx

18.

17.

d (cosh x) = sinh x dx

FIRST DERIVATIVE TEST FOR MONOTONICITY

A function f (x) is continuous at x = c if and only if the following three conditions hold:

Suppose that f is continuous on [a, b] and differentiable on (a, b).

1.

f (c) exists

(c lies in the domain of f )

If f ¿(x) > 0 at each x H (a, b), then f is increasing on [a, b].

2.

lim f (x) exists

( f has a limit as x : c )

If f ¿(x) < 0at each x H (a, b), then f is decreasing on [a, b].

limf (x) = f (c)

(the limit equals the function value)

FIRST DERIVATIVE TEST FOR LOCAL EXTREMA

3.

x:c

x:c

9 0 0 0 0

9

1

if f ¿ changes from negative to positive at c, then f has a local minimum at c;

2.

if f ¿ changes from positive to negative at c, then f has a local maximum at c;

2.

If f ¿(c) = 0 and f –(c) 7 0 , then f has a local minimum at x = c.

THE FUNDAMENTAL THEOREM OF CALCULUS If f is continuous on [a, b] then F (x) =

La

(1)

d x dF = f (t) dt = f (x), dx dx La

L a

23.

1 d (sinh-1 x) = dx 11 + x 2

24.

1 d (cosh-1 x) = dx 1x 2 - 1

25.

1 d (tanh - 1 x) = dx 1 - x2

26.

1 d (coth - 1 x) = dx 1 - x2

27.

-1 d (sech - 1 x) = dx x 11 - x 2

28.

-1 d (csch - 1 x) = dx ƒ x ƒ 11 + x 2

= - ln ƒ csc u ƒ + C

v du L

16.

L

17.

cosh u du = sinh u + C L

sinh u du = cosh u + C

u du = sin-1 Qa R + C L 1a2 - u2 1 du -1 u = a tan Qa R + C 19. L a2 + u 2

2.

k du = ku + C (any number k) L

3.

(du + dv) = du + dv L L L

20.

u 1 du = a sec-1 ` a ` + C L u 1u2 - a2

L

un du =

21.

4.

u du = sinh-1 Qa R + C L 1a2 + u2

(a 7 0)

5.

L

du u = ln ƒ u ƒ + C

22.

u du = cosh-1 Qa R + C L 1u2 - a2

(u 7 a 7 0)

6.

L

23.

L

7.

cos u du = sin u + C L

24.

x sin 2x + C cos2 x dx = + 4 2 L

8.

sec2 u du = tan u + C L

25.

sec u du = ln ƒ sec u + tan u ƒ + C L

9.

L

26.

L

27.

1 1 sec3 x dx = sec x tan x + ln ƒ sec x + tan x ƒ + C 2 2 L

10. 2

cot u du = ln ƒ sin u ƒ + C

L

If f – < 0 on I, the graph of f over I is concave down.

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L

1.

INFLECTION POINT

780321 608109

u dv = uv -

USEFUL INTEGRATION FORMULAS

SECOND DERIVATIVE TEST FOR CONCAVITY

If f –(c) = 0 and the graph of f (x) changes concavity across c then f has an inflection point at c.

csc u cot u du = -csc u + C

tan u du = - ln ƒ cos u ƒ + C L = ln ƒ sec u ƒ + C

14.

f (x) dx = F (b) - F (a).

L

L

eu du = eu + C L au 15. + C, (a 7 0, a Z 1) audu = ln a L

b

21.

If f – > 0 on I, the graph of f over I is concave up.

12.

13.

INTEGRATION BY PARTS FORMULA

1.

f (x) f ¿(x) , = lim g(x) x : a g¿(x)

a … x … b.

d (csch x) = -csch x coth x dx

if f ¿ does not change sign at c (that is, f ¿ is positive on both sides of c or negative on both sides), then f has no local extremum at c.

f (t) dt is continuous on

If f is continuous at every point of [a, b] and F is any antiderivative of f on [a, b], then

d (coth x) = - csch2 x dx

3.

11.

x

[a, b] and differentiable on (a, b) and

d (sech x) = -sech x tanh x 22. dx

20.

lim

x:a

assuming that the limit on the right side exists.

If f ¿(c) = 0 and f –(c) = 0 , then the test fails. The function f may have a local maximum, a local minimum, or neither.

3.

d (tanh x) = sech2 x dx

Suppose that c is a critical point ( f ¿ (c) = 0) of a continuous function f 2. that is differentiable in some open interval containing c, except possibly at c itself. Moving across c from left to right, 1.

If f ¿(c) = 0 and f –(c) < 0 , then f has a local maximum atx = c.

19.

Let y = f (x) be twice-differentiable on an interval I.

ISBN-13: 978-0-321-60810-9 ISBN-10: 0-321-60810-0

Suppose that f (a) = g(a) = 0 , that f and g are differentiable on an open interval I containing a, and thatg ¿(x) Z 0 on I ifx Z a . Then

1.

(2)

Applications of Derivatives Continuity

L’HÔPITAL’S RULE

Suppose f – is continuous on an open interval that containsx = c .

Integration

13.

DIFFERENTIATION RULES

x: q

lim xn =

provided the limit exists; equivalentlyf ¿(x) = lim

3. lim k = k, (k constant)

x:; q

For an integer n > 0 , lim xn = q, x:- q

Constant Multiple Rule: lim (k – f (x)) = k – L The limit of a constant times a function is the constant times the limit of the function.

5.

lim k = k,

x:c

f (x + h) - f (x) , f ¿(x) = lim h h:0

1.

SECOND DERIVATIVE TEST FOR LOCAL EXTREMA

18.

du = u + C

un+1 + C n + 1

(n Z -1)

sin u du = -cos u + C

2

csc u du = -cot u + C

sec u tan u du = sec u + C L 3

sin2 x dx =

x sin 2x + C 4 2

csc u du = - ln ƒ csc u + cot u ƒ + C

Calculus Review, Single Variable Volume of Solid of Revolution La

DISK V =

SHELL V =

LENGTH OF y = f (x)

2

p[ f (x)] dx

La

b

L =

2pxf (x) dx

La

b

L a

b

2 [ f ¿(t)]2 + [ g¿ (t)]2 dt

L =

L c

d

21 + [ g¿ (y)]2 dy

0! = 1, 1! = 1, 2! = 1 – 2, 3! = 1 – 2 – 3, n! = 1 – 2 – 3 – 4 % n

USEFUL CONVERGENT SEQUENCES 1. 2.

where x = f (t), y = g(t)

3.

Numerical Integration

4.

TRAPEZOID RULE

5.

L a

b

f (x) dx L

lim 1n = 1

¢x (y + 2y1 + 2y2 + % + 2yn- 1 + yn) 2 0

6.

lim

SLOPE OF THE CURVE r ⴝ f (u)

d 2y dx2

=

3.

xn = 0 n!

= 0 and gbn converges, then gan converges.

6.

ln(1 + x) = a n= 1

7.

(-1)n x 2n+ 1 , ƒxƒ … 1 tan - 1 x = a 2n + 1 n= 0

an = q and gbn diverges, then gan diverges. If lim n : q bn

THE RATIO TEST

an+1 an Ú 0, lim an = r , then gan converges if r 6 1 , diverges if n: q r 7 1, and the test is inconclusive ifr = 1.

Á

+ an .

(a)

an converges if lim sn exists. a n: q n= 1

(b)

an diverges if lim sn does not exist. a n: q n= 1

b

1 2 1 [ f (u)]2du r du = a 2 La 2 L

AREA OF A SURFACE OF REVOLUTION OF A POLAR CURVE (ABOUT x-AXIS) 2

q

an Ú 0, lim 1an = r, then gan converges if r 6 1, diverges if

2n

n- 1

(-1) n

x

n

, -1 6 x … 1

q

8.

q m (Binomial Series) (1 + x)m = 1 + a a b x k, ƒ x ƒ 6 1 k=1 k

m(m - 1) m m , where a b = m, a b = 1 2 2 m(m - 1) % (m - k + 1) m a b = ,k Ú 3 k! k

n

n: q

GEOMETRIC SERIES

r 7 1, and the test is inconclusive if r = 1 .

FOURIER SERIES

a ar n= 1

ALTERNATING SERIES TEST

1 2p 1 2p f (x) cos k x dx, The Fourier Series for f (x) is a0 + a (ak cos kx + bk sin kx) where a0 = f (x) dx, ak = p 2p L0 0 k=1 L q

n- 1

=

a , r 6 1, and diverges if ƒ r ƒ Ú 1. 1 - r ƒ ƒ

n: q

1 bk = p

L0

2p

f (x) sin kx dx, k = 1, 2, 3, Á

q

n= 1

THE INTEGRAL TEST Let {an} be a sequence of positive terms. Suppose thatan = f (n) , where f is a continuous, positive decreasing function for allx Ú N q (N a positive integer). Then the series g n = N an and the improper q integral 1N f (x) dx both converge or both diverge.

If g ƒ an ƒ converges, then gan converges.

TESTS FOR CONVERGENCE OF INFINITE SERIES

TAYLOR SERIES

2.

Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor Series generated by f at x = a is

3.

1.

f (k)(a) k a k! (x - a) = f (a) + f ¿(a)(x - a) + k=0

4.

q

p-SERIES 1 a n p converges if p 7 1, diverges if p … 1. n= 1 q

dr 2 S = 2pr sin u r + a b du du La B b

n

THE ROOT TEST

5. 6.

The nth-Term Test: Unless an : 0, the series diverges.

Geometric series: gar n converges if ƒ r ƒ 6 1; otherwise it diverges.

p-series: g1冫n p converges if p 7 1; otherwise it diverges.

Series with nonnegative terms: Try the Integral Test, Ratio Test, or Root Test. Try comparing to a known series with the Comparison Test or the Limit Comparison Test.

Series with some negative terms: Does g ƒ an ƒ converge? If yes, so does gan since absolute convergence implies convergence.

Alternating series: ganconverges if the series satisfies the conditions of the Alternating Series Test.

f –(a) f (n)(a) (x - a)n + % . (x - a)2 + % + n! 2!

f ¿(u) cos u - f (u) sin u

provided

an

n : q bn

q (-1) x , all x cos x = a n= 0 (2n)!

5.

ABSOLUTE CONVERGENCE TEST

2

b

d dy b a ` du dx (r, u)

If lim

q (-1) x 2n+ 1 , all x sin x = a n= 0 (2n + 1)!

= c 7 0, then gan and gbn both converge or both

an fails to exist or is different from zero. a an diverges if nlim :q

A =

CONCAVITY OF THE CURVE r ⫽ f(u)

2.

an

n : q bn

dr L = r + a b du du a B L

AREA OF REGION BETWEEN ORIGIN AND POLAR CURVE r = f(u)

provided dx/du Z 0 at (r, u).

If lim

an Ú 0, g(-1)n+ 1an converges if an is monotone decreasing and lim an = 0.

2

q

n

THE nth-TERM TEST FOR DIVERGENCE

b

dy f ¿(u) sin u + f (u) cos u ` = dx (r, u) f ¿(u) cos u - f (u) sin u

1.

diverge.

x lim a1 + n b = ex n: q n: q

q

4.

LENGTH OF POLAR CURVE

Polar Coordinates

y x = r cos u y = r sin u, x2 + y2 = r2, u = tan - 1 a x b

xn e x = a , all x n= 0 n!

n: q

q

EQUATIONS RELATING POLAR AND CARTESIAN COORDINATES

3.

gan diverges if there is a divergent series of nonnegative terms gdn with an Ú dn for all n 7 N, for some integer N.

1

q

b - a n , n is even and yi = a + i¢x, y0 = a, yn = b

1 = a (-1)nx n, ƒ x ƒ 6 1 1 + x n= 0

lim xn = 0 ( ƒ x ƒ 6 1)

q

¢x (y + 4y1 + 2y2 + 4y3 + % + 2yn- 2 + 4yn- 1 + yn) f (x) dx L 3 0 a L

2.

gan converges if there is a convergent series gcn withan … cn for all n 7 N, for some integer N.

lim x 冫n = 1 (x 7 0)

n: q

Let sn = a1 + a2 +

b

(a)

q

Suppose that an 7 0 and bn 7 0 for all n Ú N (N an integer).

SEQUENCE OF PARTIAL SUMS

SIMPSON’S RULE

1 = a x n, ƒ x ƒ 6 1 1 - x n= 0

LIMIT COMPARISON TEST

n

n: q

n

b - a where ¢x = n and yi = a + i¢ x, y0 = a, yn = b

where ¢x =

n: q

1.

(b)

ln n n = 0

lim

Let gan be a series with no negative terms.

COMPARISON TEST

FACTORIAL NOTATION

21 + [ f ¿ (x)]2 dx

LENGTH OF x = g (y)

LENGTH OF PARAMETRIC CURVE L =

USEFUL SERIES

Infinite Sequences and Series

b

If a = 0, we have a Maclaurin Series for f (x) .

dx Z 0 at (r, u). du

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