Exam Technique - C4

Question 1

Drawing modulus

Answer 1

|f(X)| - make the negative values positive
f(|x|) - make the negative X quadrants a reflection on of the positive
|f(|X|)| - as above, but reflect negative y value into positive quadrants.

Question 2

Integration by parts

Answer 2

Choose u based on u:LATE
Set up grid. Differentiate u to find du/dx
Integrate dv/dx to u. 
I=uv-[i[v(du/dx)]]
LABEL EACH PART OF THE TABLE

Repeat the integration if the second part does not integrate easily. Remember to use brackets when subbing back into original equation to ensure correct sign.

Question 3

Integration by substitution

Answer 3

This is the opposite of chain rule, hence du/dx.

Find du/dx, and rearrange to find du

Get other terms in terms of u, trying to use part of the du term where possible

Take all integers outside, but KEEP THEM WRITTEN IN ALL SUBSEQUENT LINES

Integrate the equation.

Change the limits if indefinite to be in terms of u, remember +C if indefinite.

Question 4

Simpsons rule

Answer 4

LOTS OF CARE NEEDED, EASY TO MAKE MISTAKE

Find h, by dividing the difference between limits by number of strips

Make a table, from minimum value to max, going up in steps of h

Find values using table mode, carefully rounding to at least 4 d.p.

Sub into equation h/3(ends + 4(odds) + 2(even))

Round to number of decimals given in the question.

Question 5

Mid ordinate rule.

Answer 5

Like Simpsons, lots of places to slip up

Find h

Set up first line of X value from xmin to xmax in stops of h

Find mid X values going half way between the original X values.

Use table mode to find the values corresponding to the mid X values to 4 d.p

Sub these into formula h(sum of y)

Round to number d.p given in answer.

Question 6

Solving trig equations with modified expressions for x

Answer 6

simplify the equation (normally in the previous question).

Let x = the new expression e.g. (2θ)

Solve the equation for x, giving your answer to the correct number of significant figures.

Modify the range, e.g. if 0

Question 7

Solving modulus questions

Answer 7

Solving: Find the positive and negative modulus, and solve for the value

Finding inverse: swap X and y

Rearrange for y, remembering to square root, e and on as late as possible

Swap y for f-1(X)

Question 8

Differentiation by product rule

Answer 8

label two parts U and V, and differentiate each part in a grid form. Multiply across, and add the two parts together.

Question 9

Finding normal to curve, proving root exists,

Answer 9

Having found gradient, find negative reciprocal to find gradient of normal. Find y co-ordinate by subbing in x value to original equation.

Question 10

What is the best form to keep rational expressions in to see where things cancel?

Answer 10

factorised form often cancels

Question 11

How do you find out how many solutions a quadratic has

Answer 11

b^2-4ac

Question 12

How do you factorise a cubic into linear * quadratic

Answer 12

find the linear (from factor), then find the numbers that combine to give the final terms. ^2 term easiest, then final term, then the x term.

Question 13

What should you do at the end of a simplifying question?

Answer 13

Make sure there are no factors left to cancel

Question 14

What is 1/2t written with negative indicies?

Answer 14

0.5t^-1.

Treat like a fraction. 1/2*1/t=1/2t

Question 15

What must the checked when writing partial fractions?

Answer 15

SIGNS!!!

Very easy to make sign error when simplifying coeffients

Question 16

What are the two methods to be aware of for solving partial fractions?

Answer 16

Substituting x values to eliminate variables, expand and compare co-efficients

Question 17

What does the x^2 go with for the expansion of binomials?

Answer 17

2! goes with x^2

Question 18

when modifying binomial expansion, how do you deal with adding two terms e.g. (1+x)^n and (1+2x+x^2)^n

Answer 18

Find original expansion, combine extra terms and substitute in for x. Remember brackets

Question 19

What must you do to modify a binomial expansion to add a co-efficient of x e.g. (1+x)^n to (1+ex)^n

Answer 19

Use brackets to expand

(ax)^2 = a^2x^2

Question 20

What is the equation for the differential of a parametric equation?

Answer 20

dy/dx=(dy/dt)/(dx/dt)

Question 21

What key part of the question must you check when finding an equation having found derivative?

Answer 21

Tangent, or Normal

Question 22

What type of parametric equation should you avoid rearranging to get the parameter rearranged?

Answer 22

anything involving t^2, because you end up with +-(x)^1/2, which is a mess

Question 23

For elliptical functions, how do you find the cartesian form?

Answer 23

Use the identity sin^2(x)+cos^2(x)=1. Rearrange x and y to make sin^2(x)+cos^2(x)

Question 24

What is the mnemonic for remembering in which quadrant graphs are positive?

Answer 24

All Suckers Take Chemistry, working anticlock from top right

Question 25

When solving an equation involving two trig functions, how do you solve in order not to miss solutions?

Answer 25

Get all the trig functions on one side and factorise, then solve each factor separately.

Question 26

Why can you not divide an equation through by sin(x)

Answer 26

sin(x) may = zero

Question 27

When solving a trig equation involving 2x, how must you go about it to get all the solutions?

Answer 27

Solve all values for 2x, remembering to used extended range (doubled), then divide all values by two

Question 28

When finding whether normal to parametric equation cuts the curve again, how do you do it if the cartesian form is a bitch to deal with?

Answer 28

Substitute x and y in terms of t into the equation of the normal and solve the resulting equation. If quadratic, use discriminant to show how many solutions

Question 29

If there is an identity in the form a^2-b^2, how can you factorise it?

Answer 29

(a-b)(a+b)

Then use identities to rewrite in the necessary form.

Question 30

When differentiating something difficult in the form u/v, how can you make life easier than using quotient rule?

Answer 30

Rewrite it as u*1/v.

Use product rule

Question 31

How can you check you have the correct partial fractions?

Answer 31

substitute a value into the original, and into the partial fractions, using CALC function

Question 32

How do you deal with adding weird terms e.g. going from expansion of (1+x)-2 to (1+2x-3)^-2

Answer 32

Expand the first lot, then just substitute the modified x term into the expansion.