P4 Flashcards
Binomial expansion formula
1 + nx + (n(n-1)(x)^2)/2! + (n(n-1)(n-2)(x)^2)/3!
What to do if (3 - 4x)^-3
Take out the 3 so it’s 3^-3 (1 + (-4/3 x)) ^-3 and make it + then the negative x in brackets so u don’t mess up the signs
Make the number taken out is also to the power of wtv it is
What to write after binomial expansion
+…
How to differentiate 2xy
Use product rule (term with x will get the dy/dx)
How to do differential equations
1.Differentiate each term, doing product rule for xy and adding dy/dx to terms with just y
2. Group the dy/dx terms together on one side and non dy/dx on the other
3. Factor out dy/dx and divide non terms by it to get dy/dx = ….
If integrating partial fractions
1.If has no power do 1/a ln|ax+b|
2. If has a power, do reverse chain rule or like (ax + b)^-2 integrated = ADD ONE TO POWER AND DIVIDE THEN DIVIDE BY A OF AX
3. Group the ln together using log rules
prove (a+b) > 2root ab
- Square everything and expand the (a+b) bracket
- Minus the 4ab
- Simplified to (a-b)^2 which anything squared has to be greater than 0
Cartesian equation using trig
1.Find sint cos t or wtv ones u need
2. Put them into the equation so sin2t -> (t/2)^2
If talking about water or smt and gives u dt/dx is this and ask for equation with no dt dx anything
Use differential equation steps
If asked angle between line and a vector
- Use DIRECTION VECTOR (2nd one) and the vector
- Scalar product / magnitude of each times together
- = cos x
Find area of the triangle with vectors
1/2 ab sinc
Find coordinates of 2 possible locations of E given AP = PE
- Find length of OP vector
- Compare to length of equal vector AP and find OP x by what = AP
- Then do the vector equation for the line with the lambda as the “by what” value
If integration by substitution and struggling to find u in denominator
- Factor out common factor if is …. + ……
- Find dx = and cancel if possible
- Should work to find u
With finding expressions for dx/dt using the dx/dt = dx/… * …/… * …/dt method
Make sure to include area or volume or whatever formula is used in the equation as can differentiate with respect to x or y or t
Prove intersections
- Write (i + lambda i) = (i + mew i) for xyz coords
- Set 2 up as simultaneous equations
- Check if intersect if it works with the 3rd equation
2 vectors are perpendicular
Scalar product of the 2 is 0
Scalar product
- 2 vectors
- Multiply the I values + the product of j values + product of z values
Integration by parts has -ve question in it after 2 integration rounds
“Move” the question to the question side, make it 2 integral f(x) and divide answer by 2
Rotated through 2pi integrations, revolution integration
Do pi (integrate) (fx)^2
