Equations To Remember Flashcards by Lily Pardoe

How to integrate f(ax + b)

Consider the value which would differentiate to give the original function
Workout the coefficient needed to balance the differentiated function and the function that you need

How well did you know this?

Not at all

Perfectly

What is dy/dx of e^5x-4

5e^5x-4

How well did you know this?

Not at all

Perfectly

How to integrate k(f’(x))/(f(x))

Try ln|f(x)| and differentiate to check

- Adjust any constants to match the original function

How well did you know this?

Not at all

Perfectly

How to integrate k f’(x)*f(x)^n

Try (f(x))^n+1 and differentiate to check

- Adjust any constants to match original function

How well did you know this?

Not at all

Perfectly

Cosine rule

Cos C = (a^2 +b^2 -c^2)/2ab

Or a^2 = b^2 + c^2 - 2bcCosA

How well did you know this?

Not at all

Perfectly

On a graph y=x(x+2)^2(3+x)

Where does it touch the x-axis and where does it cross the x-axis

Touches at x= -2

Crosses at x= -3

How well did you know this?

Not at all

Perfectly

What is the y-intercept of y = 4^x

How well did you know this?

Not at all

Perfectly

What is the y-intercept of y= 2e^x

How well did you know this?

Not at all

Perfectly

How to convert y = ax^n into y = mx + c

y = ax^n
Log(y) = Log(a) + Log(x^n) 
Log(y) = Log(a) + nLog(x)

How well did you know this?

Not at all

Perfectly

How to solve fg(x)

Solve g(x) to make x the subject
Place into function of f(x)

How well did you know this?

Not at all

Perfectly

Graph of y= |f(x)|

Reflect lines below x-axis above the x-axis

How well did you know this?

Not at all

Perfectly

Graph of y=f(|x|)

Mirror graph in y-axis showing reflection of positive values

How well did you know this?

Not at all

Perfectly

Nth term of arithmetic sequence

Un = a + (n-1)d

How well did you know this?

Not at all

Perfectly

Nth term of geometric sequence

Un = ar^(n-1)

How well did you know this?

Not at all

Perfectly

Sum of first n terms for an arithmetic series

Sn = n/2(2a+(n-1)d)
or
Sn = n/2(a+l)

How well did you know this?

Not at all

Perfectly

Sum of first n terms for geometric series

Sn = a(1-r^n)/1-r
or
Sn = a(r^n-1)/r-1

How well did you know this?

Not at all

Perfectly

Sum to infinity for a convergent series

S = a/1-r

How well did you know this?

Not at all

Perfectly

Binomial expansion when n is a fraction or negative

(1+x)^2 = 1 + nx + (n(n-1)/2!)x^2 + (n(n-1)(n-2)/3!)x^3

How well did you know this?

Not at all

Perfectly

How to use binomial expansion on (a+bx)^n

(a+bx)^n = (a(1+(b/a)x)^n = a^n(1+(b/a)x)^n

How well did you know this?

Not at all

Perfectly

Area of a sector in radians

(1/2)r^Ø

How well did you know this?

Not at all

Perfectly

small angle approximation of cosØ

CosØ = 1 - (ø^2)/2

How well did you know this?

Not at all

Perfectly

1 + tan^2x =

Sec^2x

How well did you know this?

Not at all

Perfectly

1 + cot^2x

Cosec^2x

How well did you know this?

Not at all

Perfectly

Sin(A+B)

sinAcosB + cosAsinB

How well did you know this?

Not at all

Perfectly

Cos(A+B)

cosAcosB - sinAsinB

Tan(A+B)

(tanA + tanB)/(1 - tanAtanB)

Sin2A

2sinAcosA

Cos2A

``` Cos^2A - Sin^2A or 2cos^2A - 1 or 1 - 2sin^2A ```

Tan2A

2tanA/1-tan^2A

asinx + bcosx can be expressed as ...

Rsin(x+ã) where Rcosã = a and Rsinã = b And R = sqrt(a^2 + b^2)

acosx + bsinx can be expressed as ...

Rcos(x-ã) where Rcosã = b and Rsinã = a And R = sqrt(a^2 + b^2)

How to convert between parametric and Cartesian equation

Solve one of the parametric equations to find x | Substitute into the other parametric equation

Domain of f(x) for parametric equations

Range of p(t) | Where x = p(t)

Range of f(x) for a parametric equation

Range of q(t) | Where y = q(t)

Chain rule:

dy/dx = dy/du x du/dx

Product rule:

uv’ + vu’

Quotient rule:

vu’-uv’/v^2

Derivative of arcsinx

1/sqrt(1-x^2)

Derivative of arccosx

-1/sqrt(1-x^2)

Derivative of arctanx

1/(1+x^2)

How to differentiate implicitly

- Differentiate x terms as normal - For terms with y, differentiate as you would with x and then multiply by dy/dx - d/dx of xy = x(dy/dx) + y - Factor out dy/dx

What does it tell you about the function if f’’(x) is < or > than 0

If f’’ < 0, function is concave | If f’’ > 0, function is convex

Newton-Raphson formula:

X(n+1) = Xn - f(Xn)/f’(Xn)

Integrate 1/x

ln|x| + c

Integrate cosecx•cotx

-cosecx + c

Integrate cosec^2x

-cotx + c

Integrate sec^2x | In formula booklet

tanx + c

Integrate secx•tanx

secx + c

Reverse chain rule: | Integrate f’(ax+b)

1/a•f(ax+b) +c

Integrate k(f’(x)/f(x))

try ln|f(x)| and differentiate to check and then adjust any constants

Integrate k(f’(x)(f(x))^n)

Try (f(x))^(n+1) and differentiate to check, and then adjust any constants

Integration by parts: | E.g, integrate uv’ (u•dv/dx)

``` uv’ = uv - integral of (vu’) u•dv/dx = uv - integral of v•du/dx ```

dy/dx = f(x)g(y) can also be written as

Integral of (1/g(y)) = integral of f(x)

Prove that sqrt(2) is irrational

``` *write assumption* then sqrt(2) = a/b And assume that a/b is a fraction in its simplest form 2 = a^2/b^2 a^2 = 2b^2 This means that a^2 is even So (2n)^2 = b^2 This means that b^2 is even If a and b are both even they have a common factor of 2. This contradicts the statement that a and b have no common factors so sqrt(2) is irrational ```

How to prove that there is an infinite number of prime numbers

*assumption* List all the prime numbers: p1, p2, p3, ..., pn Consider the number N = p1•p2•p3• ... •pn + 1 You can divide n by any of the prime numbers in the equation as you get a remainder of 1 so N has no factors. N must either be prime or have a prime factor which isn’t on the list of possible prime numbers Therefore there is an infinite number of prime numbers

What is the discriminant and what does it show about the roots of an equation?

If b^2 - (4ac) > 0, then f(x) has two roots If b^2 - (4ac) = 0, then f(x) has one repeated root If b^2 - (4ac) < 0, then f(x) has no real roots

What is the set notation for x>-2 and x<4?

{x: -2

What is the set notation for x3?

{x: x3}

What does the graph of y=k/x look like

If k>0, two curves approaching asymptotes in diagonal quadrants, (+,+) and (-,-) If k<0, two curves approaching asymptotes in diagonal quadrants, (+,-) and (-,+)

What does the graph of y=k/x^2 look like

If k>0, two curves approaching asymptotes in adjacent quadrants, all above the x-axis (conductors graph) If k<0, two curves approaching asymptotes in adjacent quadrants, all below the x-axis

If y=a^(kx), where k is real and a>0, then what is dy/dx?

dy/dx = ka^(kx)ln(a)

How to find the gradient of a point on a curve given parametric equations

- Differentiate each equation without changing into Cartesian equation. - Divide dy/dt by dx/dt to get dy/dx. - Substitute in a a value of t to get the gradient

Solving differential equations

Differentiate implicitly Factor out dy/dx Separate the variables Integrate both sides