Exam Technique - C4 Flashcards
Drawing modulus
|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.
Integration by parts
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.
Integration by substitution
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.
Simpsons rule
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.
Mid ordinate rule.
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.
Solving trig equations with modified expressions for x
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
Solving modulus questions
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)
Differentiation by product rule
label two parts U and V, and differentiate each part in a grid form. Multiply across, and add the two parts together.
Finding normal to curve, proving root exists,
Having found gradient, find negative reciprocal to find gradient of normal. Find y co-ordinate by subbing in x value to original equation.
What is the best form to keep rational expressions in to see where things cancel?
factorised form often cancels
How do you find out how many solutions a quadratic has
b^2-4ac
How do you factorise a cubic into linear * quadratic
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.
What should you do at the end of a simplifying question?
Make sure there are no factors left to cancel
What is 1/2t written with negative indicies?
0.5t^-1.
Treat like a fraction. 1/2*1/t=1/2t
What must the checked when writing partial fractions?
SIGNS!!!
Very easy to make sign error when simplifying coeffients
What are the two methods to be aware of for solving partial fractions?
Substituting x values to eliminate variables, expand and compare co-efficients
What does the x^2 go with for the expansion of binomials?
2! goes with x^2
when modifying binomial expansion, how do you deal with adding two terms e.g. (1+x)^n and (1+2x+x^2)^n
Find original expansion, combine extra terms and substitute in for x. Remember brackets
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
Use brackets to expand
(ax)^2 = a^2x^2
What is the equation for the differential of a parametric equation?
dy/dx=(dy/dt)/(dx/dt)
What key part of the question must you check when finding an equation having found derivative?
Tangent, or Normal
What type of parametric equation should you avoid rearranging to get the parameter rearranged?
anything involving t^2, because you end up with +-(x)^1/2, which is a mess
For elliptical functions, how do you find the cartesian form?
Use the identity sin^2(x)+cos^2(x)=1. Rearrange x and y to make sin^2(x)+cos^2(x)
What is the mnemonic for remembering in which quadrant graphs are positive?
All Suckers Take Chemistry, working anticlock from top right
When solving an equation involving two trig functions, how do you solve in order not to miss solutions?
Get all the trig functions on one side and factorise, then solve each factor separately.
Why can you not divide an equation through by sin(x)
sin(x) may = zero
When solving a trig equation involving 2x, how must you go about it to get all the solutions?
Solve all values for 2x, remembering to used extended range (doubled), then divide all values by two
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?
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
If there is an identity in the form a^2-b^2, how can you factorise it?
(a-b)(a+b)
Then use identities to rewrite in the necessary form.
When differentiating something difficult in the form u/v, how can you make life easier than using quotient rule?
Rewrite it as u*1/v.
Use product rule
How can you check you have the correct partial fractions?
substitute a value into the original, and into the partial fractions, using CALC function
How do you deal with adding weird terms e.g. going from expansion of (1+x)-2 to (1+2x-3)^-2
Expand the first lot, then just substitute the modified x term into the expansion.
