Unit 1 Flashcards
Partial Fractions: Denominators of the form (ax+b)(cx+d)
A/(x+1) + b/(x-2)…… (extended with arbitrary constants)
Partial Fractions: Denominators of the form (ax + b)(cx + d)^2
When we have a repeated factor we have to use both the factor and its square as denominators:
A/(x+2) + B/(x-1) + c/(x-1)^2
To find B in this situation you will have to substitute x in as 0 and use the values of C and A which you have allready found to calculate B
Partial Fractions: an irreduciable quadratic factor in the denominator
When the denominator of a partial fraction contains a quadratic factor that does not factorise - usually in the form (ax + b)(cx^2 + d) - then it will be in the form A/(ax+b) + bx+c/(cx^2+ d)
Partial Fractions: Improper Fractions
Where the numerator of a fraction is a polynomial of the same degree or higher than the denominator we have to reduce it before attempting to find the partial fractions through long division
What is the derivative of sin(ax + b)
acos(ax+b)
What is the derivative of cos(ax+b)
-sin(ax+b)
What is the chain rule equation
dy/dx = dy/du x du/dx
What are the three new trigonometric ratios
cosec x =1/sinx
secx = 1/cosx
cotx = 1/tanx
What are the exact trig values
Sine : 0 degrees= 0 30 degrees = 1/2 45 degrees = 1/root2 60 degrees = root3/2 90 degrees = 1
Cosine reverse the order
For tan divided the sin by the cos
What must be remembered when implicitly differentiating
d/dx y^2 for example:
You must use the chain rule as we are taking the derivative with respect to x and y is a function that depends on x
What are the log rules
ln(x * y) = lnx + lny
ln(x/y) = lnx-lny
lnx^r = rlnx
Find the constraint equation of a Parametirc function steps
- Find t or the other letter used in the first parametric equation
- Subsitutie the value into the second parameteric equation
- Solve for x and y on the left and the rest on the rightq
Parametric Differntiation: How to find the first derivative
- Use the equation f’(x) = y’(t)/x’(t)
Parametric Differentiation: Finding the second derivative
- Find dy/dx = dy/dt divided by dx/dt
2. Divide the derivative of dy/dx by dx/dt
Speed of a projectile given by parametric equations
speed = the root of (dx/dt squared + dy/dt squared)
The integral of sinx and cosx
sin x = -cosx
cosx = sinx
The integral of sin(ax+b)
-1/a cos (ax + b)
integral of cos(ax+b)
1/a sin (ax + b)
what is sin^2 x equal too
1/2 (1-cos2x)
what is cos^2 x equal too
1/2(1 + cos2x)
What is the integral of e^ax + b
1/a e^ax+b
What is the integral of sec^2(ax + b)
1/a tan (ax +b)
What is the integral of 1/ax+b
1/a ln ax + b
When the numerator is a derivative of the denominator, integrating the fraction will give:
lnf(x) which is essentially ln(the denominator)
Steps to Integrate by substituion
- Let u equal a variable to get rid of all x variables
- Manipulate it to cover all parts of the equation
- Now integrate the equation in the u state
- Replace the U variables with their matching variables from the original form
What is the formula for integration by parts
product = original x integrated - the integral of orginal derivative x integrated
Remeber the same variable must be integrated cant be two
When should you use a dummy variable
If a function has a standard derivastive but no standar integral then in order to integrate these a dummy variable, 1, should be introduced
What is the order of a differential equation
The order is the order of the highest derivative involed
What is the degree of a differential equation
The degree is the power of the highest derivative involved
What is a general solution
A function which satisfies the derivative is called a general solution of the differential equation
What is a particular solution
When c is found by substituting values for the rest of the variables
How to solve a first order differential equation than can be expressed in the form f(y) dy/dx = g(x)
- Seperate the variables into the form f(y) dy = g(x) dx
2. Integrate both sides
How to solve a second order linear differential equation of the form ad^2y/dx^2 + bdy/dx + cy = q(x) that are homogenous meaning q(x) = 0
- y = Ae^mx
- Substitute its derivatives into the equation
- Simplify
- Solve the subsequent auxillary equation and plug into the appropriate solution form
What solution should be used for roots that are real and distinct in the auxillary equation
y = Ae^m1x + Be^m2x
What solution should be used for auxillary roots which are real and coincident (two of the same)
y = Ae^mx + Bxe^mx
What solution should be used when the roots are complex
y = e^px(Acosqx + Bsinqx)
How to solve non homogenous ones
- General solution = complementary function(homohenous solve) + particular integral
How to find the particular integral
Find the two variables that are given as the PI form
Sub back in
How to find the form of the particular integral
Try the same form as q(x) If q(x) is a wave function (sinax or cosax) then try y= csinax + dcosax
What form should be used if there is only one root to the auxillary equation
y= A+ Be^mx
What is the chain rule formula
dy/dx = dy/du * du/dx
Trig equations
sin(2x) = sinxcosx
