Differentiation Questions Flashcards
If f is any function where x denotes ANY value in the domain of f such that f is differentiable at x, so that f does have a gradient at the point (x, f(x)), what are co-ordinates of a second point on a curved graph?
(x + h, f(x + h))
How is the gradient of a line passing through two point defined as a function and what is this known as?
f( x+h) - f( x)/h, known as the difference quotient
What happens as P2 (x+h, f(x +h)) gets closer to P1 (x, f(x))?
Value of h gets closer to zero, the value of the difference quotient gets closer to the gradient of the graph at the point of (x, (f)), thus getting closer to fâ(x)
How do you find formula of fâ(x)
You consider what happens TO the difference quotient for f AT x, as h gets closer to zero, so you need to find, in terms of x, the LIMIT of the difference quotient as h tends to zero.
What is an example of a linear graph?
Time someone has been walking and his displacement assuming vertical axis changes at a constant rate with respect to the horizontal axis
How might a curve appear in a graph?
Where velocity gradually decreases over time(where a walker slows down).
How does gradient differ between linear/curved graphs?
A linear graph has a constant rate of change whilst a curved graph has no single gradient value applying to the whole graph, instead a gradient appearing at each individual point.
How do you find the slope of a linear graph?
Slope is equal to rise/run; you can pick ANY two points on a line to determine slopes; (-3, -3) and (-1, 0), change calculated by difference of how many integers there are from P1 to P2 on both X and Y axis; X = 3 and Y =2, slope being 2/3.
What are the possible values of rise/run on a linear graph?
Positive, negative or zero, depending on the relevant co-ordinates, increasing, decreasing or being equal between two points.
What and why are the only runs without gradients?
Zero value runs, as division by zero is impossible.
What does gradient of a curved graph reference to?
A point in which, if you traced a pen along the graph, where upon reaching the marked point you just carry on moving in the direction it has been moving, it will move a straight line in whichever way you do this, being the tangent.
What is an example of a graph with no gradient?
Y = [X] where tracing a pen along the graph towards the origin, moving in the same direction upon getting to the origin would differ depending on whether you did this from right to left or left to right.
What is said of a graph with discontinuity?
It has no tangent nor gradient at that point.
