Winter Exam Flashcards
f(x) –> f(x+a)+b
Translation of (-a,b)
f(x) –> bf(ax)
Stretch 1/a parallel to x, stretch b parallel to y
In f(ax+b), which operation is done first?
The translation!
f(x) –> -f(x)
Reflection in x axis
f(x) –> f(-x)
Reflection in y axis
f^-1(x)
Inverse function, maps input onto output (basically reflection in y=x)
What makes a function even? (3)
Symmetry at y axis, all even powers with constants allowed, f(x)=f(-x)
What makes a function odd?
Rotational symmetry order 2 about origin, all odd powers and no constants, f(-x)=-f(x)
You know what?
A worthy… ;)
What would you use to differentiate a function of a function?
Chain rule, dy/dy = dy/du x du/dx.
The easy one basically
What would you use to differentiate two multiplied functions? (y=uv)
Product rule, dy/dx = v(du/dx) + u(dv/dx)
What would you use to differentiate two divided functions? (y=u/v)
Quotient rule, dy/dx = (v(du/dx) - u(dv/dx))/v^2
How do you differentiate when x is the subject?
dy/dx = 1/dx/dy
d/dx e^ax
ae^ax
d/dx lnax
1/x
d/dx ln(f(x))
f’(x)/f(x)
d/dx lnx^a
a/x
d/dx ae^x
ae^x
d/dx e^f(x)
f’(x)e^f(x)
d/dx sinkx
kcoskx
d/dx coskx
-ksinkx
1 + tan^2 x
-sec^2(x)
d/dx a^kx
ka^kx lna
Integration by substitution: 4 steps
- define u
- work out limits in terms of u
- work out du/dx and rearrange as needed
- proceed with integration!
Integral of e^ax
1/a e^ax + c
Integral of 1/x
lnx + c
Integral of f’(x)/f(x)
ln|f(x)| + c
Integral of sinax
-1/a cosax + c
Integral of cosax
1/a sinax + c
Integral of 1/ax+b
1/a ln|ax+b| + c
Integral of U(du/dx) =
UV - (int)V(du/dx) dx
sec(x)
1/cos(x)
cosec(x)
1/sin(x)
cot(x)
1/tan(x)
1+cot^2 (x)
cosec^2 (x)
1+tan^2 (x)
sec^2 (x)
sin(A+-B)
sinAcosB+-cosAsinB
cos(A+-B)
cosAcosB-+sinAsinB
tan(A+-B)
(tanA+-tanB)/(1-+tanAtanB)
cos2A
2cos^2 A - 1 and 1-2sin^2 A
For small positive x
sin(x)~~tan(x)~~(x)
cos(x)~~(1-(x)^2)/2
Guess what
You’re amazing! :)
