Everything Flashcards
What does it imply when a geometric progression CONVERGES
when the modulus of “r” is less than 1
When is a rigid body in equilibrium
When the moment about any point is 0, AND when the resultant force is 0
What is the value of the reaction force at the at all other points excluding the point around which the rod is tilting
0
At the point of inflection , the second derivative is…
0
When is a curve increasing
When the derivative is greater than zero
When is a curve decreasing
When the derivative is less than 0 or negative
When is a curve stationary
When the derivative is equal to zero
When can a section of a curve be defined as concave upwards
When the second derivative is positive(local minimum), or when the curve has an increasing gradient
When can a section of a curve be defined as concave downwards
When the second derivative is negative(local maximum), or when the gradient of the curve is decreasing
When is the point of inflection
The point When a curve changes from concave upwards to concave downwards or vice verse
At the point of inflection, what is the value of the second derivative
0
What transformation has been applied when f(x) is mapped onto f(x+a)
Translation by vector -a
0
What transformation has been applied when f(x) is mapped onto f(x)+a
Translation by vector 0
a
What transformation has been applied when f(x) is mapped onto f(ax)
Stretch by scale factor 1/a
What transformation has been applied when f(x) is mapped onto af(x)
Stretch by scale factor a
What transformation has been applied when f(x) is mapped onto -f(x)
Reflection in the x axis
What transformation has been applied when f(x) is mapped onto f(-x)
Reflection in the y axis
How do we differentiate exponential
We take natural logs of each side , then we differentiate implicitly
Derivative of tan x
Sec^2 x
Derivative of secx
Secxtanx
Derivative of cotx
-cosec^2x
Derivative of cosecx
-cosecxcotx
Derivative of any expression in the form y= Alnx
dy/dx= A/x
Integral of an expression in the form 1/ax+b
1/a ln|ax+b|+ c
What are the conditions considered when proving that root of 2 is irrational
That root 2 can be written as a/b
A and b are integers
A/b is a fraction in its simplest form, so a and b can’t both be even
Assumption made to find the time of flight
Vertical displacement is 0
Assumption made to find maximum height
Vertical final velocity is 0
Time of flight
The time that the particle is in the air
what assumptions do we make in moments when a rigid body is in equilibrium
the resultant force is zero
and the total moment about any point on the rigid body is zero
both assumptions are required for a rigid body to be in equilibrium
when an object is uniform what does it mean for the center of mass
it is directly in the middle of the object
during tilting, apart from the point around which the object is being tilted, what will be the value of the reaction force of all other points
0
At limiting equilibrium, what is the formula of friction
F= uR
At limiting equilibrium, acceleration is
0
At rest, what is the formula for friction
F<= uR
when estimating the area under a curve using rectangles, how do we ensure that our approximation is more accurate?
we increase the number of strips of rectangles
In numerical integration, what is the formula for finding h using the trapezium rule
 Difference between the limits/ number of strips
Concave downwards is aka
Concave
Concave upwards is aka
Convex
Ordinate is aka
The y-coordinate
In (1,2) , 2 is the ordinate as it is the y-co-ordinate
The general formula for the trapezium rule
0.5h (1st term+last term+ 2(rest of the terms))
How do we find the number of strips using the number of ordinates (trapezium rule)
Num of ordinates - 1
Cases in which the change of method sign may fail
When the curve touches the y-axis
When the curve has a vertical asymptote
When they are several roots in the interval
Formula for the Length or magnitude of a 3D vector
√(a2 + b2 + c2).
The 2s represent “squared”
The range of validity of (1+x)^n
The modulus of x should be less than 1
Domain
Where x can exist
Range
The value of y that the graph or function reaches
What is the rule governing the reaction force when a rod is at the point of tilting
When at the point of tilting , the reaction force around any other point but the point at which the rod is tilting is 0
What does it imply if something is modelled as a rod
It is treated as a rigid, one dimensional object
If a particle is projected horizontally, then what would be the value of the vertical component of the initial velocity
0
Why does speed increase when velocity and acceleration are negative
If velocity and acceleration are negative, then the object is moving in the opposite direction of its positive position. If the object continues to move in the opposite direction, then the velocity and acceleration will become more negative, which means that the speed will increase.
Why does speed increase when velocity decreases
If the velocity is decreasing, then the object is slowing down. However, if the object is still moving in the same direction, then the speed will still increase. This is because the speed is the magnitude of the velocity, so even if the velocity is decreasing, the speed can still increase if the velocity remains in the same direction.
To prove that a certain line is a tangent to a circle , what needs to be done?
Substitute the equation of the line into the equation of the circle
Solve the equation that forms to show that it has only one real root
This will indicate that the line only meets the circle at one point , so it’s a tangent to the circle
How do you check whether two lines are perpendicular
Their gradients multiply together to give -1
How is an even function represented
Example
F(-x)= f(x)
Y=cosx
When the transformations are all in the x axis, then what is the order we apply then in
Translation, then stretch
When the transformations are all in the y axis, what is the order we apply them in
Stretch
Then translation
Odd functions represented by
Example
F(-x)= -f(x)
Y=sinx
Arithmetic sequences nth term formula
Un =a + (n-1)d
Geometric sequence nth teen formula
Un= ar^n-1
Pie radians in degrees
180 degrees
Small angle approximation for cosx
Cosx= 1- x^2/2
When the shaded area of a curve is rotated At right angles or a right angle to the x axis , what happens
A solid with a volume is formed
To find the volume of a solid formed by rotating a shaded area of a curve about the x-axis, we can use integration. We need to integrate the function that defines the curve from the lower limit to the upper limit of the shaded area, and then multiply the result by 2π.
to converge to a root, what is the rule (numerical methods)
The gradient of the curve should be between -1 and 1
How to find resultant vectors
Add the vectors together
Position vector
Tells us how to get from the origin to the coordinate
Cosec Has vertical asymptotes when y is equal to?
0
