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Calc > Calc 1 > Flashcards

Calc 1 Flashcards

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)

A

1/x

22
Q

derivative of a^x

A

ln(a) * a^x

23
Q

derivative of e^x

A

e^x !!!

24
Q

derivative of log(small a)(of x)

A

= 1 / (ln(a)*x)

25
Q

y’ =

A

dy/dx

26
Q

f’(x) =

A

1 / g’(f(x))

27
Q

d/dx[sin^-1x] = 1/(sqrt(1-x^2))

A
28
Q

d/dx[cos^-1(x)] =

A

-1/[sqrt(1-x^2)]

29
Q

d/dx[tan^-1(x)] =

A

1/[1 + x^2]

30
Q

d/dx[tan^-1(x)] =

A

1/[1 + x^2]

31
Q

ln(a^b) = b * ln(a)

A
32
Q

y = b^x, 0<b<1

A

limb^x as x approaches inf. = 0
limb^x as x approaches -inf. = inf.

33
Q

y = b^x, b > 1

A

limb^x as x approaches inf. = inf.
limb^x as x approaches - inf. = 0

34
Q

f(x) = a^x is multiplicative over any interval of length k with factor a^k:

A

f(y + k) = a^k * f(y)

35
Q

e^ln6 =

A

6

36
Q

/inf. = 0??

A
37
Q

Log of a constant seems to be 0

A

Also, ln(10x) = ln(10) + ln(x)

38
Q

s(u) = ln f(u)

A

s’(u) = f’(u) / f(u)

f’(u) = f(u) * der. of ln f(u)

39
Q

ln (x/y) = lnx - lny

A
40
Q

ln(xy) = lnx + lny

A
41
Q

limit of ratio = ratio of limits if

A

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
Q

What if (see previous)

L and M = 0
L = +/- inf. and M = +/- inf.
L is not 0 and M = 0

A

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
Q

Indeterminate powers

A

0^0; 1^inf. ; inf.^0

44
Q

differentiablilty implies continuity

A
45
Q

concave upward

A

smile… slope increasing

f’(x) increasing
f’‘(x) > 0

46
Q

concave downward

A

frown… slope decreasing

f’(x) decreasing
f’‘(x) < 0

47
Q

derivative of anti derivative

A

= the function!!

48
Q

To find general antiderivqtive, just

A

Add C to your antiderivative

49
Q

Summation notation for Riemann sums…

A

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
Q

distance = velocity * (change in time)

A

think about the units and plotting on a graph and how you can use riemann sums!!!

51
Q

Definite integral =

A

Limit of Reimann sum

Calc (3 decks)

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