Equations To Remember Flashcards
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
What is dy/dx of e^5x-4
5e^5x-4
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 to integrate k f’(x)*f(x)^n
- Try (f(x))^n+1 and differentiate to check
- Adjust any constants to match original function
Cosine rule
Cos C = (a^2 +b^2 -c^2)/2ab
Or a^2 = b^2 + c^2 - 2bcCosA
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
What is the y-intercept of y = 4^x
1
What is the y-intercept of y= 2e^x
2
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 to solve fg(x)
Solve g(x) to make x the subject Place into function of f(x)
Graph of y= |f(x)|
Reflect lines below x-axis above the x-axis
Graph of y=f(|x|)
Mirror graph in y-axis showing reflection of positive values
Nth term of arithmetic sequence
Un = a + (n-1)d
Nth term of geometric sequence
Un = ar^(n-1)
Sum of first n terms for an arithmetic series
Sn = n/2(2a+(n-1)d)
or
Sn = n/2(a+l)
Sum of first n terms for geometric series
Sn = a(1-r^n)/1-r
or
Sn = a(r^n-1)/r-1
Sum to infinity for a convergent series
S = a/1-r
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 to use binomial expansion on (a+bx)^n
(a+bx)^n = (a(1+(b/a)x)^n = a^n(1+(b/a)x)^n
Area of a sector in radians
(1/2)r^Ø
small angle approximation of cosØ
CosØ = 1 - (ø^2)/2
1 + tan^2x =
Sec^2x
1 + cot^2x
Cosec^2x
Sin(A+B)
sinAcosB + cosAsinB
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
