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Calculus Flashcards

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1
Q

Define a function.

A

A function f is the correspondence between two set the domain and codomain, that assigns each element of the domain too one and only one element of C

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2
Q

Define range.

A

The set of all images is called the range of f

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3
Q

Define the graph of a function.

A

The graph of a function f is the set of points(x,y) in the xy-plane with x ∈ Domf and y = f(x)

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4
Q

What is the vertical line test.

A

If any vertical line intersects the curve more than once then the curve is not the graph of a function, otherwise it is

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5
Q

Define an even function.

A

A function f is even if f(x) = f(-x) ∀ x ∈ Domf

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6
Q

Define an odd function.

A

A function f is odd if f(x) = -f(x) ∀ x ∈ Domf

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7
Q

What is the equation for an odd function?

A

fodd(x) = ½ f(x) – ½ f(-x)

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8
Q

What is the equation for an even function?

A

feven(x) = ½ f(x) + ½ f(-x)

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9
Q

Define a step function.

A

A step function is a piecewise function which is constant on each piece.

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10
Q

Define surjective.

A

A function f : D ⟼ C is surjective (or onto) if Ranf = C

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11
Q

Define injective.

A

A function f : D ⟼ C is injective (or one-to-one) if ∀ x1, x2 ∈ D with x1 ≠ x2 then f(x1) ≠ f(x2)

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12
Q

What is the horizontal line test.

A

If any horizontal line intersects the graph of f more than once then f is not injective, otherwise it is

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13
Q

Define bijective.

A

A function f : D ⟼ C is bijective if it is both surjective and injective

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14
Q

What is the theorem of inverse functions?

A

A bijective function f has a unique inverse f−1 defined by f−1(f(x)) = x = f(f−1(x)) or (f ◦ f−1)(x) = x = (f−1 ◦ f)(x)

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15
Q

Define periodic.

A

A function f(x) is periodic if ∃ p > 0 s.t. f(x+p) = f(x) ∀ x

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16
Q

Define a limit.

A

f(x) has a limit L as x tends to a if: ∀ ℇ > 0 s.t. |f(x) - L| < ℇ when 0 < |x - a| < δ. We then write:
limx→a f(x) = L or equivalently f(x) → L as x → a. If there is no such L then we say that no limits exists

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17
Q

Define continuous at a point.

A

A function f(x) is continuous at the point x = a if the following three properties hold:
• f(a) exists
• limx→a f(x) exists
• limx→a f(x) is equal to f(a)

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18
Q

Define continuous on a subset.

A

A function f(x) is continuous on a subset S of its domain if it is continuous at every point in S.

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19
Q

Define continuous.

A

A function f(x) is continuous if it is continuous at every point in its domain.

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20
Q

What is trig 1?

A

limx→0 (sinx)/x = 1

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21
Q

What is trig 2?

A

limx→0 (1-cosx)/x = 0

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22
Q

Define the limit as x → ∞.

A

f(x) has a limit L as x → ∞ if: ∀ ℇ > 0 ∃ S > 0 s.t. |f(x) - L| < ℇ when x > S. We then write limx→∞ f(x) = L or equivalently f(x) → L as x → ∞. If there is no such L then we say that no limit exists.

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23
Q

Define a right-sided limit.

A

f(x) has a right-sided limit L+ = limx→a+ f(x) as x tends to a from above if: ∀ ℇ > 0 ∃ δ > 0 s.t. |f(x) - L+| < ℇ when 0 < x - a < δ

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24
Q

Define a left-sided limit.

A

f(x) has a left-sided limit L− = limx→a− f(x) as x tends to a from below if: ∀ ℇ > 0 ∃ δ > 0 s.t. |f(x) - L−| < ℇ when 0 < a - x < δ

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25
What is removable discontinuity?
In this case L exists but f(a) ≠ L, this type of discontinuity can be removed to make a continuous function (hence the name removable).
26
What is a jump discontinuity?
In this case both L+ and L− exist but L+ ≠ L−
27
What is an infinite discontinuity?
In this case at least one of L+ or L− does not exist
28
What is the intermediate value theorem?
The intermediate value theorem states that if f(x) is continuous on [a,b] and u is any number between f(a) and f(b) then ∃ c ∈ (a,b) s.t. f(x) = u.
29
What is the equation for the definition of the derivative as a limit?
f’(a) = limh→0 (f(a+h)-f(a))/h
30
What is the formula for Euler's way to differentiate?
Picture
31
What is the chain rule theorem?
The chain rule theorem states that if g(x) is differentiable at x and f(x) is differentiable at g(x) then thecomposition f ◦ g(x) is differentiable at x with: f ◦ g’(x) = f’(g(x))g’(x)
32
What is L'Hopital's rule?
If f(x) and g(x) are differentiable on I (interval) = (a - h, a) ∪ (a, a + h) for some h > 0, with limx→a f(x) = limx→a g(x) = 0. If limx→a exists and g’(x) ≠ 0 ∀ x ∈ I then: limx→a (f(x))/(g(x)) = limx→a (f'(x))/(g'(x))
33
Define an upper bound.
If ∃ a constant k1 s.t. f(x) ≤ k1 ∀ x in I we say that f(x) is bounded above in I and we call k1 an upper bound of f(x) in I.
34
Define global maximum value.
If ∃ x ∈ I s.t. f(x1) = k1, then we say that the bound is attained, and we call k1, the global maximum value of f(x) in I.
35
Define lower bound.
If ∃ a constant k2 s.t. f(x) ≥ k2 ∀ x in I we say that f(x) is bounded below in I and we call k2 a lower bound of f(x) in I.
36
Define global minimum value.
If ∃ x ∈ I s.t. f(x2) = k2, then we say that the bound is attained, and we call k2, the global minimum value of f(x) in I.
37
Define bounded.
f(x) is bounded in K if it is both bounded above and bounded below in I. i.e If ∃ a constant k s.t. |f(x)| ≤ k ∀ x in i.
38
What is the extreme value theorem?
The extreme value states that if f is a continuous function on a closed interval [a,b] then it is bounded on that interval and has upper and lower bounds that are attained: ∃ points x1 and x2 in [a,b] s.t. f(x2) ≤ f(x) ≤ f(x1) ∀ x ∈ [a,b]
39
Define monotonic increasing.
f(x) is monotonic increasing in [a,b] if f(x1) ≤ f(x2) ∀ x1, x2 with a ≤ x1 < x2 ≤ b.
40
Define strictly monotonic increasing.
f(x) is strictly monotonic increasing in [a,b] if f(x1) < f(x2) ∀ x1, x2 with a ≤ x1 < x2 ≤ b.
41
Define local maximum.
We say that f(x) has a local maximum at x = a if ∃ h > 0 s.t. f(a) ≥ f(x) ∀ x ∈ (a - h, a + h).
42
Define local minimum.
We say that f(x) has a local minimum at x = a if ∃ h > 0 s.t. f(x) ≥ f(a) ∀ x ∈ (a - h, a + h).
43
Define stationary point.
f(x) has a stationary point at x = a if it is differentiable at x = a with f’(a) = 0
44
Define critical point.
Need to see online notes
45
Define endpoint maximum.
If c is an endpoint of f(x) then f has an endpoint maximum at x = c if f(x) ≤ f(c) for x sufficiently close to c.
46
What is Rolle's theorem?
Rolle’s theorems states that if f is differentiable on the open interval (a,b) and continuous on the closed interval [a,b], with f(a) = f(b), then there is at least one x ∈ (a,b) for which f’(c) = 0.
47
What is a corollary to Rolle's theorem?
If f(x) is differentiable on an open interval I, then any roots of f(x) are separated by a root of f’(c) = 0
48
What is the mean value theorem?
The mean value theorem states that if f is differentiable on the open interval (a,b) and continuous on the closed interval [a,b], then there is at least one c ∈ (a,b) for which: f’(c) = (f(b)-f(a))/(b-a)
49
Define an indefinite integral.
F(x) is an indefinite integral of f(x) in the interval (a,b) if F’(X) = f(x) ∀ x ∈ (a,b) (also called the anti-derivative)
50
Define integrable in.
We say that f(x) is integrable in (a, b) if it has an indefinite integral F(x) in (a, b) that is continuous in [a,b]
51
Define the Riemann sum.
Suppose f(x) is a function defined for x ∈[a,b]. The Riemann sum is .....picture
52
Define definite integral.
The definite integral ba∫f(x).dx = limh→0 R (the Riemann sum)
53
What is the fundamental theorem of calculus?
If f(x) is continuous on [a, b] then the function F (x) = xa∫f(t).dt defined for x ∈ [a, b] is continuous on [a, b] and differentiable on (a,b) and is an indefinite integral of f(x) on (a, b). F’(x) = d/dx xa∫f(t) .dt = f(x) throughout (a,b).
54
What is the equation for the fundamental theorem of calculus?
F(x) = t2t1∫f(t).dt , then F’(x) = f(t)(d(t2))/dx - f(t)(d(t1))/dx
55
What is the integral of, picture
arctan(x) + c
56
What is the integral of, picture
arcsin(x) + c
57
What is the integral of, picture
arcsin(x) + c
58
What is the integral of, picture
-cot(x) + c
59
What is the integral of an odd function on a symmetric interval?
0
60
What is the interval of an even function on a symmetric interval?
2 a0∫feven(x).dx
61
Define order.
The order of an ODE is the highest number of derivatives that occur.
62
What change of variable do you usually use in a first order homogeneous ODE?
y = xv
63
What are the four types of first oder ODEs?
1. ) separable 2. ) homogeneous 3. ) linear 4. ) exact
64
What is the general form for first order separable ODEs?
dy/dx = f(x)f(y)
65
How do you solve first separable order ODEs?
Picture
66
What is the general form of first order homogeneous ODES?
f(tx, ty) = f(x,y)
67
How do you solve first oder homogeneous ODEs?
Use the substitution y = xv, and then solve it as a a separable ODE in terms off v and x
68
What is the general form of first order linear ODEs?
f(x,y) = -p(x)y + q(x)
69
How do you solve first order linear ODES?
Picture
70
What is the general form of first order exact ODEs?
M(x, y) dx + N(x, y) dy = 0
71
How do you show if an equation is exact?
if ∂M/∂x= ∂N/∂x
72
What do you have to do if the ODE isn't exact?
multiply through by an integration factor so that now ∂m/∂x= ∂n/∂x where m=IM and n=IN
73
What is a Bernoulli equation?
nonlinear ODE of the form y’ + p(x)y = q(x)yn
74
How did you solve Bernoulli ODEs?
Use the substitution v = y^(1-n) and then solve as a linear ODE for v(x)
75
For second order ODEs, what is the general solution when you have distinct real roots?
Picture
76
For second order ODEs, what is the general solution when you have repeated real roots>
Picture
77
For second order ODEs, what is the general solution for complex roots?
Picture
78
Whats Taylors theorem?
Taylors theorem states that if f(x) has n + 1 continuous derivatives in an open interval I that contains the point x = a; then ∀ x ∈ I, picture
79
What is the equation for the Taylor polynomial?
Picture
80
Define o(xn)
Let n be a positive integer. We say that f(x) = o(xn) (as x → 0) if limx→0 (f(x) )/x^n)= 0
81
What is the equation for Fourier series?
Picture
82
What is the equation for an in Fourier series?
Picture
83
What is the equation for bn in Fourier series?
Picture
84
What does cos(n∏) equal?
(-1)^n
85
What is Dirichlet's theorem?
Let f(x) be a periodic function, with period 2L; such that on the interval (- L, L) it has a finite number of extreme values, a finite number of jump discontinuities and |f(x)| is integrable on (-L, L): Then its Fourier series converges for all values of x. Furthermore, it converges to f(x) at all points where f(x) is continuous and if x = a is a jump discontinuity then it converges to: picture
86
What is Persevak's theorem?
If f(x) is a function of period 2L with Fourier coefficients an, bn then picture
87
What is the equation for the Taylor polynomial when we have two variables?
Picture
88
What is the equation for the remainder of the taylor polynomial when we have two variables?
Picture
89
Define y simple.
A region D of the plane is called y simple if every line that is parallel to the y-axis and intersects D, does so in a single line segment (or a single point if this is on the boundary of D).
90
Define x simple.
A region D of the plane is called x simple if every line that is parallel to the x-axis and intersects D, does so in a single line segment (or a single point if this is on the boundary of D).
91
Define the Jacobian transformation.
Picture

Calculus (1 decks)

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