Fun Calculus Definitions and Theorems – Compiled by Henry Qin from Mathworld and the Hughes-Hallet Single Variable Calculus Intermediate Value Theorem Suppose f is continuous on a closed interval [a, b] . If k is any number between f ( a ) and f ( b ) , then there is at least one number c in [a, b] such that f ( c ) = k . Definition of Limit
We define lim f ( x ) to be the number L (if one exists) such that for every ε > 0 (as small as we
x →c
want), there exists a δ > 0 (sufficiently small) such that if x − c < δ and x ≠ c, then f ( x ) − L < ε . Examples: Page 50-51
Definition of Continuity
The function f is continuous at x = c if f is defined at x = c and if lim f ( x ) = f ( c ) In other words, f ( x ) is as close as we want to f ( c ) provided x is close enough to c . The function is
continuous on an interval [ a, b ] if it is continuous at every point on the interval.
x →c
(If c is an endpoint of the interval, we define continuity at x = c using one-sided limits at c.)
Definition of Derivative
The derivative of f at a, written f ' ( a ) , is defined as
f (a + h) − f (a) . h →0 h If the limit exists, then f is said to be differentiable at a. f ' ( a ) = lim
Definition of Derivative Function
For any function f, we define the derivative function, f ' , by f ( x + h) − f ( x) . f ' ( x ) = Rate of change of f at x = lim h→0 h
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Definition of Differentiability
1. 2. 3. 4.
The function is continuous. The limit exists The function exists. The limit equals the function.
A Differentiable Function is Continuous If f ( x ) is differentiable at a point x = a , then f ( x ) is continuous at x = a .
Section 3.6, Page 139 shows fun uses of the chain rule
Derivative of General Inverse Function (with proof) d f ( f −1 ( x ) ) = 1 dx d ⇒ f ' ( f −1 ( x ) ) ⋅ ( f −1 ( x ) ) = 1 dx d 1 ⇒ ( f −1 ( x ) ) = dx f ' ( f −1 ( x ) )
(
)
Hyperbolic Functions (used in engineering) e x + e− x cosh x = 2
e x − e− x sinh x = 2
Section 3.8, Page 146 – properties
cosh 2 x − sinh 2 x = 1
d ( cosh x ) = sinh x dx
d ( sinh x ) = cosh x dx
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The Tangent Line Approximation (local linearization)
Suppose f is differentiable at a. Then, for values of x near a, the tangent line approximation to f ( x ) is
f ( x ) ≈ f ( a ) + f ' ( a )( x − a )
The expression f ( a ) + f ' ( a )( x − a ) is called the local linearization of f near x = a . We are thinking of a as fixed, so that f ( a ) and f ' ( a ) are constant. The error, E ( x ) , in the approximation is defined by
E ( x ) = f ( x ) − f ( a ) − f ' ( a )( x − a ) .
Differentiability and Local Linearity
Suppose f is differentiable at x = a and E ( x ) is the error in the tangent line approximation, that is:
E ( x ) = f ( x ) − f ( a ) − f ' ( a )( x − a ) .
Then lim
x →a
E ( x) = 0. x−a
The Mean Value Theorem If f is continuous on a ≤ x ≤ b and differentiable on a < x < b , then there exists a number c, with a < c < b , such that f (b) − f ( a ) f '(c) = b−a In other words, f ( b ) − f ( a ) = f ' ( c )( b − a )
The Increasing Function Theorem
We say that f is increasing on an interval if, for any two numbers x1 and x2 in the interval such that
x1 < x2 , we have f ( x1 ) < f ( x2 ) . If instead we have f ( x1 ) ≤ f ( x2 ) , we say that f is nondecreasing.
Suppose that f is continuous on a ≤ x ≤ b and differentiable on a < x < b . • If f ' ( x ) > 0 on a < x < b then f is increasing on a ≤ x ≤ b . • If f ' ( x ) ≥ 0 on a < x < b , then f is nondecreasing on a ≤ x ≤ b .
The Constant Function Theorem
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Suppose that f is continuous on a ≤ x ≤ b and differentiable on a < x < b . If f ' ( x ) = 0 on a < x < b , then f is constant on a ≤ x ≤ b .
The Racetrack Principle
Suppose that g and h are continuous on a ≤ x ≤ b and differentiable on a < x < b , and that g ' ( x ) ≤ h ' ( x ) for a < x < b . • • If g ( a ) = h ( a ) , then g ( x ) ≤ h ( x ) for a ≤ x ≤ b . If g ( b ) = h ( b ) , then g ( x ) ≥ h ( x ) for a ≤ x ≤ b .
Definition of Critical Points For any function f, a point p in the domain of f where f ' ( p ) = 0 or f ' ( p ) is undefined is called a
critical point of the function. In addition, the point ( p, f ( p ) ) on the graph of f is also called a critical
point. A critical value of f is the value, f ( p ) , at a critical point, p.
Local Extrema and Critical Points
Suppose f is defined on an interval and has a local maximum or minimum at the point x = a , which is not an endpoint of the interval. If f is differentiable at x = a , then f ' ( a ) = 0 . Thus, a is a critical points.
The Second-Derivative Test for Local Maxima and Minima
• • •
If f ' ( p ) = 0 and f " ( p ) > 0 then f has a local minimum at p. If f ' ( p ) = 0 and f " ( p ) < 0 then f has a local maximum at p. If f ' ( p ) = 0 and f " ( p ) = 0 then the test tells us nothing.
A point at which the graph of a function changes concavity is called an inflection point of f. - Potential inflection points are located where f " = 0 and f " is undefined. These are the places where it can change signs To find the global maximum and minimum of a continuous function on a closed interval, a ≤ x ≤ b : Compare values of the function at all candidate points: the critical points in the interval and the endpoints.
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The Extreme Value Theorem
If f is continuous on the closed interval a ≤ x ≤ b , then f has a global maximum and a global minimum on that interval.
L’Hopital’s Rule If f and g are differentiable and f ( a ) = g ( a ) = 0 , then
lim
x →a
f ( x) f '( x) = lim , g ( x ) x→a g ' ( x )
provided the limit on the right exists.
L’Hopital’s rule applies to limits involving infinity, provided f and g are differentiable: • When lim x →a f ( x ) = ±∞ and lim x →a g ( x ) = ±∞ or
•
When a = ±∞ and lim f ( x ) = lim g ( x ) = 0 or lim f ( x ) = ±∞ and lim g ( x ) = ±∞ .
x →∞ x →∞ x →∞ x →∞
It can be shown that under these circumstances: f ( x) f '( x) lim = lim x →a g ( x ) x→a g ' ( x ) (where a may be ±∞ ), provided the limit on the right-hand side exists.
Dominance
We say that g dominates f as x → ∞ if lim
x →∞
f ( x) = 0. g ( x)
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Left- and right-hand Riemann sums
Left-hand sum =
∑ f ( t ) Δt
i =0 i
n −1
Right-hand sum =
∑ f ( t ) Δt
i =1 i
n
Taking the Limit to Obtain the Definite Integral
Suppose f is continuous for a ≤ t ≤ b . The definite integral of f from a to b, written Is the limit of the left-hand or right-hand sums with n subdivisions of a ≤ t ≤ b as n gets arbitrarily large. In other words, b ⎛ n ⎞ f ( t ) dt = lim ⎜ ∑ f ( ti ) Δt ⎟ ∫a n →∞ ⎝ i =1 ⎠ and b ⎛ n −1 ⎞ ∫a f ( t ) dt = lim ⎜ ∑ f ( ti ) Δt ⎟ . n →∞ ⎝ i =0 ⎠ Each of these sums is called a Riemann sum, f is called the integrand, and a and b are called the limits of integration.
∫ f ( t ) dt ,
b a
The (First) Fundamental Theorem of Calculus
If f is continuous on the interval [ a, b ] and f ( t ) = F ' ( t ) , then
∫ f ( t ) dt = F ( t )
b a
b a
= F (b) − F ( a )
The Second Fundamental Theory of Calculus (Construction Theorem for Antiderivatives)
If f is a continuous function on an interval, and if a is any number in that interval, then the function F defined as follows is an antiderivative of f:
F ( x ) = ∫ f ( t ) dt
x a
Using the Construction Theorem for Antiderivatives x sin t F ( x) = ∫ dt = Si ( x ) 0 t
Method of Partial Fractions
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We split a factored denominator into two fractions, each of whose integrals is easy to evaluate. 1 1 ⌠ ⎞ ⌠ ⎛ − 13 + 3 ⎟ dx dx = ⎮ ⎜ ⎮ ⌡⎝ x−2 x−5⎠ ⌡ ( x − 2 )( x − 5 ) 1 A B = + ( x − 2 )( x − 5) x − 2 x − 5 1 = A ( x − 5) + B ( x − 2 ) = ( A + B ) x − 5 A − 2 B Equating Coefficients
−5 A − 2 B = 1 A+ B = 0 1 1 ⇒ A= − ,B = 3 3
-
See page 331-333 for special cases
Q( x ) P( x )
Strategy for Integrating a Rational Function,
• •
If degree of P ( x ) ≥ degree of Q ( x ) , try long division and the method of partial fractions on the remainder. If Q ( x ) is the product of distinct linear factors, use partial fractions of the form
A . ( x − c)
•
If Q ( x ) contains a repeated linear factor, ( x − c ) , use partial fraction of the form
n
An A1 A2 + + ... + n 2 ( x − c ) ( x − 2) ( x − c) • If Q ( x ) contains an unfactorable quadratic q ( x ) , try a partial fraction of the form
Ax + B q ( x)
Trigonometric Substitutions
To simplify
a 2 − x 2 , for constant a, try x = a sin θ , with −
π
2
≤θ ≤
π
2
To simplify a 2 + x 2 or
a 2 + x 2 , for constant a, try x = a tan θ with −
π
2
<θ <
π
2
Diverging and Converging Improper Integrals
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Suppose f ( x ) is positive for x ≥ a . If lim ∫ f ( x ) dx is a finite number, we say that
b b →∞ a
∞
∫ f ( x ) dx converges and define
a b b
∞
∫ f ( x ) dx = lim ∫ f ( x ) dx Otherwise, we say that ∫ f ( x ) dx diverges. We define ∫ f ( x ) dx similarly.
a b →∞ a
∞
a
−∞
Splitting Integrals with Two Infinite Limits We can use any (finite) number c to define
∫
∞
−∞
f ( x ) dx = ∫
c
−∞
f ( x ) dx + ∫ f ( x ) dx
c
∞
If either of the two new improper integrals diverges, we say the original integral diverges. Only if both of the new integrals have a finite value do we add the values to get a finite value of the original integral.
Another Type of Improper Integral: When the Integrand Becomes Infinite Suppose f ( x ) is positive and continuous on a ≤ x ≤ b and tends to infinity as x → b .
If lim ∫ f ( x ) dx is a finite number, we say that −
c c →b a b a c →b
−
∫ f ( x ) dx
b a c a
converges and define
∫ f ( x ) dx = lim ∫ f ( x ) dx
Otherwise, we say that
∫ f ( x ) dx
b a
diverges.
Suppose f ( x ) is positive and continuous on [ a, b ] except at point c. If f ( x ) tends to infinity as
x → c , then we define
∫
b
a
f ( x ) dx = ∫ f ( x ) dx + ∫ f ( x ) dx .
c b a c
If either of the two new improper integrals diverges, we say the original integral diverges. Only if both of the new integrals have a finite value do we add the values to get a finite value of the original integral.
Useful Integrals for Comparison ∞ ⌠ 1 dx converges for p > 1 and diverges for p ≤ 1 ⎮ p ⌡1 x
1 ⌠ 1 dx converges for p > 1 and diverges for p ≤ 1 ⎮ p ⌡0 x
∫
∞
0
e − ax dx converges for a > 0
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