PPQ Corrections 3

Question 1

Z with a line over the top =

Answer 1

The conjugate of Z (change the sign).

Question 2

Don’t forget to ________ maclaurin expansion.

Answer 2

simplify

Question 3

Proof by counterexample

Answer 3

prove the opposite is true

Question 4

Volume of revolution

Answer 4

X-axis= ∫πy^2
Y-axis= ∫πx^2

Question 5

How to prove if numbers are in a sequence

Answer 5

Find a common difference (d for arithmetic and r for geometric)

Question 6

Un (arithmetic) =

Answer 6

a+(n-1)d

Question 7

Un (geometric) =

Answer 7

ar^(n-1)

Question 8

First order linear differential equations.

Answer 8

1. Integrate the part that is with the y (P(x)).
2. Put step one as e^.
3. Multiply both sides by step 2.
4. Make LHS d/dx.(y.step 2)
5. Integrate both sides and find y (LHS is just what’s in the bracket at step 5).

Question 9

What happens when e^lnx

Answer 9

e^ln cancels and you are left with x.

Question 10

Proof by induction steps:

Answer 10

1. Goal
2. Let n=1 (LHS=RHS)
3. Assume true for n=k
4. Prove true for n=k+1 (use step 3)
5. Statement

Question 11

Inverse matrix

Answer 11

1/detA (d -b)
(-c a)

Question 12

Don’t forget __ when integrating

Answer 12

+ C

Question 13

Displacement=

Answer 13

∫ velocity

Question 14

Velocity=

Answer 14

∫ acceleration or ds/dt

Question 15

Acceleration=

Answer 15

dv/dt

Question 16

How to find slant asymtote

Answer 16

Carry out division
As x –> infinity
remainder –> 0
then y –> answer without remainder

Question 17

If the degree of the denominator is greater than the degree of the numerator then the horizontal asymtote =

Answer 17

0

Question 17

What happens to a graph when you are asked to draw lf(x)I ?

Answer 17

Everything is above the x-axis.

Question 18

What would be the PI for 35e^2x ?

Answer 18

Cxe^2x

Question 19

A x Identity matrix =

Answer 19

A

Question 20

I^2=

Answer 20

I

Question 21

2A divided by A = ?
(matrices)

Answer 21

2I

Question 22

When does a sum to infinity exist in a geometric sequence?

Answer 22

When r < 1

Question 23

sum to infinity in geometric

Answer 23

a/1-r

Question 24

sum to infinity in arithmetic sequence

Answer 24

always undefined

Question 25

Why do 2x3 matrices not have an inverse?

Answer 25

They are not square matrices. Only square matrices have an inverse.

Question 26

Rate of change =

Answer 26

d(what you're trying to find)/dt

Question 27

If an equation is x = a

Answer 27

The gradient is undefined (denominator of gradient = 0).

Question 28

How would you find a matrix that has an anticlockwise rotation (A) FOLLOWED BY a reflection along the x-axis (B)

Answer 28

BxA in that order.

Question 29

Intersection of 3 planes: A line of intersection.

Answer 29

Redundant equation on bottom of matrix. Let z=t. Sub in and find x and y.

Question 30

Angle between 2 planes

Answer 30

Find a.b where a and b are equations of 2 planes. Find √x^2+y^2+z^2 for a and b. Find cosθ= a.b/IaI IbI Final answer must be ACUTE angle.

Question 31

At the end of logarithmic differentiation...

Answer 31

Substitute y for it's x term (what you initially differentiated).

Question 32

Odd functions have...

Answer 32

Half turn symmetry

Question 33

Contrapositive statement

Answer 33

The contrapositive statement is true therefore the original statement is true.

Question 34

Parametric equation (vectors)

Answer 34

t = (x-x1)/a = (y-y1)/b = (z-z1)/c

Question 35

How to find parametric equations of a line passing through two points.

Answer 35

Find direction vector. State parametric equations.

Question 36

Equation of a plane using 3 points.

Answer 36

1.Find direction vectors A->B and A->C. 2. Find A->B x A->C 3. Step2 . A (constant point) 4. Write equation (step 2) = (step 3)

Question 37

Proof by induction statement.

Answer 37

Since statement is true for n=1 assumed true for n=k and by assumption true for n=k+1 then statement is true for all n∈N.

Question 38

Polar form:

Answer 38

r (cosθ + isinθ)

Question 39

Proof by counterexample

Answer 39

Prove original statement is false.

Question 40

How to find the intersection of two lines.

Answer 40

Set parametric equations equal to each other and find values λ and t using simultaneous equations.

Question 41

Function is even when

Answer 41

f(x) = f(-x)

Question 42

Function is odd when

Answer 42

f(x) = -f(x)

Question 43

Vector equations =

Answer 43

( ) ( ) (point) + t (direction vector) ( ) ( )

Question 44

How to show that two lines intersect.

Answer 44

Check z component in both parametric equations are equal. z1= z2

Question 45

Equation of a plane using two lines.

Answer 45

1. L1 x L2 2. (L1xL2) . a point on either line. 3. Write equation: Step 1 = Step 2.

Question 46

Differential equation when roots are not real

Answer 46

y=e^(r)x (Ccos(i)x + Dsin(i)x) r=real i=imaginary

Question 47

How to prove something is a root.

Answer 47

Sub it in and if it is a root the eqn will = 0.

Question 48

How to find inverse of 3x3 matrix.

Answer 48

Set up (A l I ). Turn into ( I l A-1) using ERO's.

Question 49

How to find instantaneous speed and acceleration.

Answer 49

Speed= magnitude using x' and y’ Acceleration= magnitude using x'' and y''

Question 50

How to find direction of speed and acceleration.

Answer 50

Using tanθ=y/x For speed use= x' and y' For acceleration use x'' and y''