Unit 1

Question 1

Partial Fractions: Denominators of the form (ax+b)(cx+d)

Answer 1

A/(x+1) + b/(x-2)…… (extended with arbitrary constants)

Question 2

Partial Fractions: Denominators of the form (ax + b)(cx + d)^2

Answer 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

Question 3

Partial Fractions: an irreduciable quadratic factor in the denominator

Answer 3

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)

Question 4

Partial Fractions: Improper Fractions

Answer 4

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

Question 5

What is the derivative of sin(ax + b)

Answer 5

acos(ax+b)

Question 6

What is the derivative of cos(ax+b)

Answer 6

-sin(ax+b)

Question 7

What is the chain rule equation

Answer 7

dy/dx = dy/du x du/dx

Question 8

What are the three new trigonometric ratios

Answer 8

cosec x =1/sinx
secx = 1/cosx
cotx = 1/tanx

Question 9

What are the exact trig values

Answer 9

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

Question 10

What must be remembered when implicitly differentiating

Answer 10

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

Question 11

What are the log rules

Answer 11

ln(x * y) = lnx + lny
ln(x/y) = lnx-lny
lnx^r = rlnx

Question 12

Find the constraint equation of a Parametirc function steps

Answer 12

1. Find t or the other letter used in the first parametric equation
2. Subsitutie the value into the second parameteric equation
3. Solve for x and y on the left and the rest on the rightq

Question 13

Parametric Differntiation: How to find the first derivative

Answer 13

1. Use the equation f’(x) = y’(t)/x’(t)

Question 14

Parametric Differentiation: Finding the second derivative

Answer 14

1. Find dy/dx = dy/dt divided by dx/dt

2. Divide the derivative of dy/dx by dx/dt

Question 15

Speed of a projectile given by parametric equations

Answer 15

speed = the root of (dx/dt squared + dy/dt squared)

Question 16

The integral of sinx and cosx

Answer 16

sin x = -cosx

cosx = sinx

Question 17

The integral of sin(ax+b)

Answer 17

-1/a cos (ax + b)

Question 18

integral of cos(ax+b)

Answer 18

1/a sin (ax + b)

Question 19

what is sin^2 x equal too

Answer 19

1/2 (1-cos2x)

Question 20

what is cos^2 x equal too

Answer 20

1/2(1 + cos2x)

Question 21

What is the integral of e^ax + b

Answer 21

1/a e^ax+b

Question 22

What is the integral of sec^2(ax + b)

Answer 22

1/a tan (ax +b)

Question 23

What is the integral of 1/ax+b

Answer 23

1/a ln ax + b

Question 24

When the numerator is a derivative of the denominator, integrating the fraction will give:

Answer 24

lnf(x) which is essentially ln(the denominator)

Question 25

Steps to Integrate by substituion

Answer 25

1. Let u equal a variable to get rid of all x variables
2. Manipulate it to cover all parts of the equation
3. Now integrate the equation in the u state
4. Replace the U variables with their matching variables from the original form

Question 26

What is the formula for integration by parts

Answer 26

product = original x integrated - the integral of orginal derivative x integrated

Remeber the same variable must be integrated cant be two

Question 27

When should you use a dummy variable

Answer 27

If a function has a standard derivastive but no standar integral then in order to integrate these a dummy variable, 1, should be introduced

Question 28

What is the order of a differential equation

Answer 28

The order is the order of the highest derivative involed

Question 29

What is the degree of a differential equation

Answer 29

The degree is the power of the highest derivative involved

Question 30

What is a general solution

Answer 30

A function which satisfies the derivative is called a general solution of the differential equation

Question 31

What is a particular solution

Answer 31

When c is found by substituting values for the rest of the variables

Question 32

How to solve a first order differential equation than can be expressed in the form f(y) dy/dx = g(x)

Answer 32

1. Seperate the variables into the form f(y) dy = g(x) dx

2. Integrate both sides

Question 33

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

Answer 33

1. y = Ae^mx
2. Substitute its derivatives into the equation
3. Simplify
4. Solve the subsequent auxillary equation and plug into the appropriate solution form

Question 34

What solution should be used for roots that are real and distinct in the auxillary equation

Answer 34

y = Ae^m1x + Be^m2x

Question 35

What solution should be used for auxillary roots which are real and coincident (two of the same)

Answer 35

y = Ae^mx + Bxe^mx

Question 36

What solution should be used when the roots are complex

Answer 36

y = e^px(Acosqx + Bsinqx)

Question 37

How to solve non homogenous ones

Answer 37

1. General solution = complementary function(homohenous solve) + particular integral

Question 38

How to find the particular integral

Answer 38

Find the two variables that are given as the PI form

Sub back in

Question 39

How to find the form of the particular integral

Answer 39

Try the same form as q(x)
If q(x) is a wave function (sinax or cosax) then try y= csinax + dcosax

Question 40

What form should be used if there is only one root to the auxillary equation

Answer 40

y= A+ Be^mx

Question 41

What is the chain rule formula

Answer 41

dy/dx = dy/du * du/dx

Question 42

Trig equations

Answer 42

sin(2x) = sinxcosx