Calculus Flashcards
In circle dr is
tiny difference in radius
You have a problem which can be approximated by the sum may very small values (dr). So the smaller, the better approximation and approaches the area under the graph.
Figure out how far a car has travelled based on its velocity at each point in time
- You might range though many different points in time, and each one multiplied by the velocity at that time times a tiny change in time, dt => v(t)dt => gives the corresponding distance travelled during that little time.
Area under parabola - x ** 2
fix the left to 0 and right vary => x. Find a function A(x) that gives the area under the parabola between 0 and x.
A(x) is called _______ of x ** 2
Integral
Finding the area under the curve
integral
When you slightly increase the x by tiny nudge -> dx -> look at the resulting change in area, sliver of rectangle, tiny difference in area-> dA.
That rectangles height is x**2 and width is dx.
Under curve the more dx -> 0, the more that sliver looks like a rectangle.
dA = x2.dx
dA/dx = x2 (Ratio of tiny change in A to the tiny change in x that caused it is approximately x**2)
We still don’t know A(x). But we do know that A(x) must have a property that, when we look at two near by points, 3, 3.0001, consider the change change to A at those two points. A(3.0001) - A(3). That difference divided by the change in input values 3.0001 -3 = 0.0001 should be equal to the value of x**2
A(3.0001) - A(3)/0.0001 = x**2
derivative of A with respect to x
If slight nudge in the output of A -> d(A) divided by a slight nudge to the input that caused it is about equal to the height of the graph at that point (f(x)).
dA/dx ~ f(x). The gets better as dx -> 0
How sensitive a function is to smaller changes to x
What is the fundamental theorem of calculus
The derivative of a function for the area under a graph gives you back the function defining the graph itself.
