Rules To Remember Flashcards
When integrating f ’(x) / f (x) what do you get?
ln |f (x) | + c
When separating partial fractions with an x² how would the A, B, C fractions be separated?
A/(x+1) + (Bx+C)/(x²+1)
What are the 4 steps for proof by induction?
- Prove that the statement holds for one value (usually n=1)
- Make an assumption for the statement that it holds for some future value (n=k)
- Use the assumption to prove that when n=k+1 the conjecture is also true
- Make a carefully worded conclusion explaining that the conjecture is true for any value of n
How would you make the second derivative ( d² y/ dx ²)?
d/dt (dy/dx) x dt/dx
How do you integrate to logarithms?
If the numerator of a fraction is the differentiated version of the denominator you can jump straight to a logarithm (ln|…|+C )
When reversing the chain rule for integral calculus:
f ‘ (x) f(x) dx=..?
card potentially needs editing
1/2 (f(x)) ² + c
In general, what is the formula for the nth term of an arithmetic series?
Un = a + (n - 1) d
When given the graph of f (x) what would the graph of |f (x)| look like?
Any parts of the graph that fall below the x-axis (into negative y-coordinates) are reflected back onto the positive side.
What type of graph has rotational symmetry (half turn symmetry) about the origin?
Odd
Which type of function has line symetry (with the y-axis being the axis of symmetry)?
An Even function
What type of functions don’t have symmetry?
These are said to be neither odd nor even functions
How do the powers of polynomials dictate whether a function is odd or even?
Functions containing only even powers of x are even and functions containing only odd powers are odd
When a function is given without a graph how do you check whether it’s odd or even?
For odd functions f (-x) = - f (x)
For even functions f(-x) = f (x)
Functions that are neither won’t return to an expression that can be written in terms of x
How do you find an objects speed from given points (equations of x and y)?
|v| =
square root of ( (dx/dt) ² + (dy/dt) ²)
How do you find the volume of revolution?
Integrate pi • (x/y) ² in respect to whatever equation you do not have and substitute the equation into the (x/y) so all terms same
What is the formula for the nth term of a geometric series?
Un = ar ^(n-1)
What is the equation for a geometric series that’s Sn goes to infinity?
S(infinity) = a / (1-r) Where - 1 < r<1
What is the formula for integrating by parts?
Integration symbol u v’ = u v - integration symbol u’ v
How would you go about solving a differential equation in the form:
dy/dx + P(x)y = Q(x) ?
- Find an integrating factor
μ(x) = e ^ (∫ P(x) dx) - Multiply both sides of equation by IF
- Integrate to find a general solution in the form
μ(x) y= ∫ μ(x) Q(x) dx
In seconds order differential equations what complimentary function would you use if the AE has two distinct roots?
y = Ae ^ (D1x) + Be ^ (D2x)
Where D1 and D2 are roots
What is the auxiliary equation AE?
a D² + b D + c = 0
Which complementary function do you use if AE has equal roots?
y = (A + Bx) e ^ (Dx)
Where D is the repeated root
What complementary function would you use if AE has no real roots?
The roots will be in the form
D = a +/ - bi
So the CF is
y = e ^ (ax) (Acosbx + Bsinbx)
State the quadratic formula?
x = ( - b +- √( b² - 4ac) ) / 2a
