Calculus 1 Final Exam Review
Wednesday 12/10/03
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Calculus 1
FINAL EXAM REVIEW
(Final Exam is Monday 12/15/03 at 1:30 PM in room 382)
Limits
Some Limits
(1.2) lim cos x  cos a
(1.2) lim c  c
xa
xa
(1.2) lim x  a
(1.2) lim e x  e a
xa
xa
(1.2) lim x  a
n
(1.2) lim ln x  ln a
n
xa
x a
(1.2) lim p( x)  p(a)
1
 0 (t > 0)
x  x t
(1.4) lim p n ( x)  
(1.4) lim
xa
(1.2) lim sin x  sin a
xa
x 
Some Properties of Limits
(1.2) lim c  f ( x)  c  lim f ( x)
(1.2) lim n f ( x)  n lim f ( x)
xa
xa
xa
(1.2)
lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
xa
x a
xa
xa
(1.2) If f(x) ≤ g(x) ≤ h(x)
and lim f ( x)  lim h( x)  L
x a
xa
then lim g ( x)  L
xa
(Squeeze Theorem)


(1.2) lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
(1.3) lim f g x   f lim g x 
f ( x)
 f ( x)  lim
xa

(1.2) lim 
xa g ( x) 
g ( x)

 lim
xa
 f ( x) 
f ' ( x)
(3.1, 7.6) lim 
for
 lim
x a g ( x) 
x a g ' ( x)


indeterminate forms 00 or 
(L’Hopital’s Rule)
xa
x a

xa
(1.2) lim  f ( x)  lim f ( x)
n
x a
x a

xa
xa
n
(1.1) A limit exists if and only if both one sided limits exist.
(1.3) f(x) continuous at x = a (i) f(a) is defined, (ii)  lim f ( x) , and (iii)
x a
lim f ( x)  f (a )
xa
(1.3) Intermediate Value Theorem: If f is continuous on [a, b] and W is a number such
that f(a) < W < f(b), then c  a, b such that f(c) = W.
(1.3) Corollary of IVT: If f is continuous on [a, b] and f(a) and f(b) have opposite signs,
then there is at least one number c (a, b) for which f(c) = 0.
(1.3) Know the method of bisections for finding zeros of a function.
(1.4) Know how to handle lim and how to find slant asymptote.
x
Calculus 1 Final Exam Review
Wednesday 12/10/03
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(1.5) Limit Definition: lim f ( x)  L if given any number ε > 0 there exists another
x a
number δ > 0 such that 0 < |x – a| < δ guarantees that |f(x) - L| < ε. (Also: lim f ( x)   if
xa
given any number M > 0 there exists another number δ > 0 such that 0 < |x – a| < δ
guarantees that f(x) > M.) (Also: lim f ( x)  L if given any number ε > 0 there exists
x 
another number M > 0 such that x > M guarantees that |f(x) - L| < ε.)
Differentiation
Definitions of Derivative
f x  h   f x 
f c  h   f c 
f  x   f c 
 lim
(2.1) f '  x   lim
(2.1) f ' c   lim
h 0
h 0
x c
h
h
xc
(2.3)
(2.3)
(2.5)
(2.5)
(2.5)
(2.5)
(2.5)
(2.5)
Some Derivatives
d
1
(6.8)
csc 1 x 
dx
x x2 1
d
c  0
dx
 
d r
x  rx r 1
dx
d
sin x  cos x
dx
d
cos x   sin x
dx
d
tan x  sec 2 x
dx
d
cot x   csc 2 x
dx
d
sec x  sec x tan x
dx
d
csc x   csc x cot x
dx
(2.6, 6.3)
 
d x
e  ex
dx
a
x
 e x ln a
(2.6, 6.1)
d
ln x  1  log b x  ln x 
dx
x 
ln b 
d
1
(6.8)
sin 1 x 
dx
1 x2


(6.8)
(6.9)
(6.9)
(6.9)
(6.9)
(6.9)
(6.9)





d
1
cot 1 x 
dx
1 x2
d
cosh x  sinh x
dx
d
sinh x  cosh x
dx
d
tanh x  sec h 2 x
dx
d
sec hx   sec hx tanh x
dx
d
csc hx   csc hx coth x
dx
d
coth x   csc h 2 x
dx






d
1
sinh 1 x 
dx
1 x2
d
1
cosh 1 x 
(6.9)
dx
x2  1
(6.9)
(6.9)
d
1
tanh 1 x 
dx
1 x2
Calculus 1 Final Exam Review






Wednesday 12/10/03
d
1
cos 1 x 
dx
1 x2
d
1
tan 1 x 
(6.8)
dx
1 x2
(6.8)
(6.8)
(2.3)
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





d
sec h 1 x 
dx
x
d
(6.9)
csc h 1 x 
dx
x
(6.9)
d
1
sec 1 x 
dx
x x2 1
(6.9)
1
1 x2
1
1 x2
d
1
coth 1 x 
dx
1 x2
Some Properties of Derivatives
d  f  x   f '  x g  x   f  x g '  x 

(2.4)
dx  g  x  
g x 2
d
cf x   cf ' x 
dx
d
 f x   g x   f ' x   g ' x 
dx
d
 f x g x   f ' x g x   f x g ' x 
(2.4)
dx
(2.3)
d
 f g x   f ' g x   g ' x 
dx
d 1
1
f x  
(6.2)
dx
f ' f 1 x 
(2.7)




(2.9) Mean Value Theorem: If f(x) is continuous on [a,b] and differentiable on (a,b),
f b   f a 
then there exists a number c  (a,b) such that f ' c  
ba
(3.1) Linear Approximation to f(x) at x = x0 is L(x) = f(x0) + f '(x0)(x - x0)
f ( xn )
(3.2) Newton’s Method: to find a zero of a function: xn1  xn 
f ' ( xn )
(3.3, 3.4, 3.5) Know the relationship between extrema, derivatives, increasing/decreasing,
concavity, inflection points, and second derivatives.
(3.7) Optimization is taking the model function for a real-world situation and finding the
exrema of that function.
Integration
Definitions of the Integral
d
F ( x)  c   f ( x), then  f ( x)dx  F ( x)  c
(4.1) if
dx
b
d
(4.5) if
F ( x)  f ( x), then  f ( x)dx  F (b)  F (a) (Fundamental Thm. Of Calc.)
dx
a
d
(4.5) if
F ( x)  f ( x), then
dx
b
(4.4)
n
x
 f (t )dt  F ( x) (Fundamental Thm. Of Calc.)
a
 f ( x)dx  lim  f (c )x, where x 
a
n
i 1
i
ba
, and ci  xi 1 , xi 
n
Calculus 1 Final Exam Review
Wednesday 12/10/03
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Some Integrals
Read Derivatives table backwards to get
 cot xdx  ln sin x  c
antiderivatives…
 tan xdx  ln sec x  c
 sec xdx  ln sec x  tan x  c
 csc xdx  ln csc x  cot x  c
 tanh xdx  ln cosh x  c
Some Properties of Integrals
f ( x)dx   g ( x)dx
(4.1)
  f x   g x dx  
(4.4)  af ( x)dx  a  f ( x)dx
(4.4)
(4.4)
b
c
b
a
a
c
 f ( x)dx   f ( x)dx   f ( x)dx for c  a, b
 f (ax) 
1
a
F (ax)  c
du
(4.6)
 f (u ) dx dx F (u )  c
(4.6)

f ' ( x)
dx  ln f ( x)  c
f ( x)
(4.4)
b
b
a
a
 g ( x)dx   f ( x)dx when g ( x)  f ( x) for all x  a, b
b
1
(4.4) Integral Mean Value Theorem: c  a, b  s.t. f (c) 
f ( x)dx
b  a a
(4.7) Know how to evaluate left-, mid-, and right-point Riemann sums.
(4.7) Know how to use fnInt( ) on your calculators.
(5.2) Volume by slices: V   A( x)dx (or V   A( y )dy ) ; for solids of rotation,


A( x)  r 2 ( x) for discs and A( x)   ro ( x)  ri ( x) for washers.
2
2
(5.3) Volume by cylindrical shells for solids of rotation: V   2r ( x)h( x)dx
b
(5.4) Arc length: s   1   f ' x  dx
2
a
b
(5.4) Surface Area for solids of rotation about y = 0: S   2f ( x) 1   f ' x  dx
2
a
(5.5) Projectile Motion:
x(t )  v0 x t  x0
y(t )   12 gt 2  v0 y t  y0
v x (t )  v0 x
v y (t )   gt  v0 y
Calculus 1 Final Exam Review
Wednesday 12/10/03
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b
(5.7) Probability: If f(x) is a pdf on [a, b], then
 f ( x)dx  1, P(c  X  d) =
a
d
b
c
c
a
a
 f ( x)dx  1,    xf ( x)dx , and the median is found by solving for c: 0.5   f ( x)dx .
(6.4) Exponential growth/decay problems arise from the differential equation
y’(t) = ky(t). The solution is always y(t) = Aekt, where k is called the growth (k > 0) or
decay (k < 0) constant.
(6.4) Relationship between half life (T½) or doubling time (T2) and k:
T½ = (ln 2) / k; T2 = (ln 2) / k
(6.4) Value of an investment compounded continuously: V(t) = $Pert.
dy
 f ( x)  g ( y ) has
(6.4, 6.5) A separable first-order differential equation y ' ( x) 
dx
y'
solution 
dx   f ( x)dx
g ( y)
(6.6) Euler’s Method: y( xi 1 )  yi 1  yi  hf ( xi , yi )
Miscellanea
(6.1) and (6.3) Know the properties of exponentials and logarithms.
(6.2) A function f has an inverse iff it is one-to-one.
(6.2) A function and its inverse are symmetric about the line y = x.
sinh x
1
sech x 
(6.9) sinh x  12 e x  e  x  cosh x  12 e x  e  x  tanh x 
cosh x
cosh x
2
2
(6.9) cosh x  sinh x  1
1 1 x
(6.0) Know formulas for inverse hyperbolic functions… i.e., tanh 1 x  ln
2 1 x