π β’ Module 12 : Differentiation & Vectors Flashcards
This lesson is about distance-time & speed-time graphs, direct & inverse proportion, gradients & differentiation, tangents & turning points, min/max problems, vectors, and conditional probability.
How do you calculate speed from a distance-time graph?
Speed Is the gradient of the graph:
Speed = Distance/Time
How do you find acceleration from a speed-time graph?
Acceleration is the gradient of the graph:
Acceleration = Change in Speed/Time
What does the area under a speed-time graph represent?
The total distance traveled.
How do you find the gradient of a curve at a specific point?
Draw a tangent at that point and calculate its gradient.
What is the derivative of f(π₯)=aπ₯βΏ?
fβ(π₯)=n x aπ₯βΏβ»ΒΉ
What do the derivatives of a function tell us?
The gradient of the function at any given point.
How do you find the equation of a tangent to a curve?
1 β Differentiate the function to find fβ(π₯).
2 β Plug in the π₯-value to get the gradient at that point.
3 β Use y=mπ₯+c to find the equation.
How do you find maximum and minimum points of a function?
1 β Differentiate the function
2 β Set fβ(π₯)=0 to find critical points.
3 β Use the second derivative test:
If fββ(π₯) > 0, itβs a minimum.
If fββ(π₯) < 0, itβs a maximum.
What is a position vector?
A vector that represents the position of a point relative to the origin.
Example: r =
[x
y]
How do you add and subtract vectors?
Add or subtract their components separately.
Addition: a + b
Subtraction: a - b
What is the dy/dx rule for differentiation?
The derivative of y=axβΏ is found using:
dy/dπ₯ = n x π₯βΏβ»ΒΉ
What is the difference between collinear, parallel, and equal vectors?
Collinear vectors: Lie on the same line or a scalar multiple of each other.
Parallel vectors: Have the same direction but may have different magnitudes.
Equal vectors: Have the same magnitude and direction, regardless of the position.
How do you determine if three points are collinear?
Find the vectors between the points and check if they are parallel (scalar multiples).
Example : If A(1,2), B(3,6), C(5,10), then:
AB = (3-1, 6-2) = (2,4)
BC = (5-3, 10-6) = (2,4)
Since AB and BC are equal, A, B, and C are collinear.
What is direct proportion and how is it written algebraically?
Two values are directly proportional if one increases when the other increases.
Itβs written as:
y=kx
Where k is a constant.
Example : If y=3x, then y triples when x triples.
How do you find the probability of A given B in a probability tree?
Use the formula:
P(A|B) = (P(A β© B)/P(B)
P(A β© B) β Probability of both A and B happening.
P(B) β Probability of B happening.
