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Calc 1 Flashcards

(51 cards)

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1
Q

A function f(x) has a removable discontinuity at x = a if

A
  1. f is not defined/not continuous at x = a
  2. f(a) could be defined/redefined so that new function is continuous at x = a
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2
Q

a function f(x) is said to have a jump discontinuity at x = a if

A
  1. the limit from both sides exists
  2. the left and right side limits are not equal
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3
Q

Studying Continuity, find:

A

Domain
Largest Intervals (right & left continous, continous)
Discontinuity
(limits, classify, remove if possible)

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4
Q

need limit symbol until…

A

you plug in 0 for h

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5
Q

Power Rule

A

Derivative of f(x) = x^n IS nx^(n-1) for any real number n

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6
Q

Constant Rule

A

Derivative of f(x) times constant = Derivative of (c * f(x))

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7
Q

Sum and Difference Rules

A

If f(x) and g(x) are both differentiatiable, then:

Derivative of [f(x) +/- g(x)] = Derivative of f(x) +/- Derivative of g(x)

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8
Q

Derivative of a constant is

A

0

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9
Q

normal line

A

perpendicular to tangent line

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10
Q

lim as x approaches 0 of (cosx - 1)/x

A

is 0

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11
Q

lim as x appr. 0 of sinx/x

A

= 1

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12
Q

derivative of the sin(x)

A

= cos(x)

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13
Q

derivative of cos(x)

A

= -sin(x)

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14
Q

typically as you take derivatives

A

functions get simpler…

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15
Q

velocity is first derivative of position, s

A

acceleration derivative of velocity

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16
Q

momentum of a moving object is

A

the product of the objects mass and velocity

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17
Q

Newton

A

Kg * (m/s^2)

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18
Q

Product Rule

A

If f and g are both differentiatable, then the derivative of f(x)g(x) = f(x)derivative of g(x) + g(x)*derivative of f(x)

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19
Q

Quotient Rule

A

If f and g are both differentiable, then the derivative of [f(x)/g(x)] = [g(x)derivativef(x) - f(x)derivativeg(x)] / g(x)^2

Derivative of numerator has to come before derivative of denominator
Denominator is squared on bottom

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20
Q

derivative of 1/g(x)

A
  • derivativeg(x)/g^2(x)
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21
Q

Derivative of ln(x)

22
Q

derivative of a^x

23
Q

derivative of e^x

24
Q

derivative of log(small a)(of x)

Study These Flashcards
A

= 1 / (ln(a)*x)

25
y' =
dy/dx
26
f'(x) =
1 / g'(f(x))
27
d/dx[sin^-1x] = 1/(sqrt(1-x^2))
28
d/dx[cos^-1(x)] =
-1/[sqrt(1-x^2)]
29
d/dx[tan^-1(x)] =
1/[1 + x^2]
30
d/dx[tan^-1(x)] =
1/[1 + x^2]
31
ln(a^b) = b * ln(a)
32
y = b^x, 0
limb^x as x approaches inf. = 0 limb^x as x approaches -inf. = inf.
33
y = b^x, b > 1
limb^x as x approaches inf. = inf. limb^x as x approaches - inf. = 0
34
f(x) = a^x is multiplicative over any interval of length k with factor a^k:
f(y + k) = a^k * f(y)
35
e^ln6 =
6
36
#/inf. = 0??
37
Log of a constant seems to be 0
Also, ln(10x) = ln(10) + ln(x)
38
s(u) = ln f(u)
s'(u) = f'(u) / f(u) f'(u) = f(u) * der. of ln f(u)
39
ln (x/y) = lnx - lny
40
ln(xy) = lnx + lny
41
limit of ratio = ratio of limits if
Limits are finite and one on bottom is not 0 Top limit infinity and bottom > 0 OR top limit -infinity and bottom < 0 (+inf) If signs dont match... -inf. If top limit is finite and bottom limit is +/- inf., ghen ratio of limits is 0 These are all determinate forms.
42
What if (see previous) L and M = 0 L = +/- inf. and M = +/- inf. L is not 0 and M = 0
0/0 and inf/inf -- see L'Hopital's Rule L/0:, ratio of limits does not exist unless bottom function does not equal 0 for all x near a... leads to essential discontinuity If bottom function greater than 0 for all x near a, limit is pos or neg inf. (Depends on sign of L) When g is always neg. for all x near a, limit is also pos or neg inf. (Sign opposite L) (THINK!) IF SIGN X CHANGES NEAR A, THERE IS NO LIMIT All three cases: Limit may or may not exist
43
Indeterminate powers
0^0; 1^inf. ; inf.^0
44
differentiablilty implies continuity
45
concave upward
smile... slope increasing f'(x) increasing f''(x) > 0
46
concave downward
frown... slope decreasing f'(x) decreasing f''(x) < 0
47
derivative of anti derivative
= the function!!
48
To find general antiderivqtive, just
Add C to your antiderivative
49
Summation notation for Riemann sums...
If the funct. is non-negative and continous in the interval, the Area under funct. Graph and above x axis is A = limas n approaches infinity (rest of summation notation) (27:30) This the definite integral
50
distance = velocity * (change in time)
think about the units and plotting on a graph and how you can use riemann sums!!!
51
Definite integral =
Limit of Reimann sum

Calc (3 decks)

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