Winter Exam Flashcards

1
Q

f(x) –> f(x+a)+b

A

Translation of (-a,b)

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2
Q

f(x) –> bf(ax)

A

Stretch 1/a parallel to x, stretch b parallel to y

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3
Q

In f(ax+b), which operation is done first?

A

The translation!

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4
Q

f(x) –> -f(x)

A

Reflection in x axis

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5
Q

f(x) –> f(-x)

A

Reflection in y axis

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6
Q

f^-1(x)

A

Inverse function, maps input onto output (basically reflection in y=x)

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7
Q

What makes a function even? (3)

A

Symmetry at y axis, all even powers with constants allowed, f(x)=f(-x)

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8
Q

What makes a function odd?

A

Rotational symmetry order 2 about origin, all odd powers and no constants, f(-x)=-f(x)

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9
Q

You know what?

A

A worthy… ;)

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10
Q

What would you use to differentiate a function of a function?

A

Chain rule, dy/dy = dy/du x du/dx.
The easy one basically

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11
Q

What would you use to differentiate two multiplied functions? (y=uv)

A

Product rule, dy/dx = v(du/dx) + u(dv/dx)

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12
Q

What would you use to differentiate two divided functions? (y=u/v)

A

Quotient rule, dy/dx = (v(du/dx) - u(dv/dx))/v^2

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13
Q

How do you differentiate when x is the subject?

A

dy/dx = 1/dx/dy

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14
Q

d/dx e^ax

A

ae^ax

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15
Q

d/dx lnax

A

1/x

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16
Q

d/dx ln(f(x))

A

f’(x)/f(x)

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17
Q

d/dx lnx^a

18
Q

d/dx ae^x

19
Q

d/dx e^f(x)

A

f’(x)e^f(x)

20
Q

d/dx sinkx

21
Q

d/dx coskx

22
Q

1 + tan^2 x

23
Q

d/dx a^kx

24
Q

Integration by substitution: 4 steps

A
  1. define u
  2. work out limits in terms of u
  3. work out du/dx and rearrange as needed
  4. proceed with integration!
25
Integral of e^ax
1/a e^ax + c
26
Integral of 1/x
lnx + c
27
Integral of f'(x)/f(x)
ln|f(x)| + c
28
Integral of sinax
-1/a cosax + c
29
Integral of cosax
1/a sinax + c
30
Integral of 1/ax+b
1/a ln|ax+b| + c
31
Integral of U(du/dx) =
UV - (int)V(du/dx) dx
32
sec(x)
1/cos(x)
33
cosec(x)
1/sin(x)
34
cot(x)
1/tan(x)
35
1+cot^2 (x)
cosec^2 (x)
36
1+tan^2 (x)
sec^2 (x)
37
sin(A+-B)
sinAcosB+-cosAsinB
38
cos(A+-B)
cosAcosB-+sinAsinB
39
tan(A+-B)
(tanA+-tanB)/(1-+tanAtanB)
40
cos2A
2cos^2 A - 1 and 1-2sin^2 A
41
For small positive x
sin(x)~~tan(x)~~(x) cos(x)~~(1-(x)^2)/2
42
Guess what
You're amazing! :)