The fundamental theorem of calculus

The Fundamental Theorem of Calculus

Suppose  f  is continuous on  [a,b].

1)    If  y=∫xaf(t)dt, then  dydx=f(x).

2)  ∫baf(t)dt=F(b)−F(a), where  F  is an antiderivative of  f.

The theorem can be applied in many instances. One of which is as shown, for finding a function that increases or is concaved upwards.

The function  F  is increasing on the interval(s) where its derivative  f  is positive in value. 

The function  F  is concave up on the interval(s) where its derivative  f  is increasing. 

    Fundamental theorem of calculus: 

    Swapping the bounds for definite integral: 

Approximating instantaneous rate of change word problem

This can be done by calculating the average of two slopes.


    Approximating instantaneous rate of change word problem:

Vertical distance of bouncing ball

It is a tricky problem from Khan Academy but seems to be worth looking at.

Since the ball travels up the same distance as it travels down (except for the first bounce), the total vertical motion is

10+12(10)+12(10)+(12)2(10)+(12)2(10)+...

=10+5+5+(12)(5)+(12)(5)+(12)2(5)+(12)2(5)+...

After the first term, the expression we have is twice the sum of an infinite geometric series with first term a=5 and common ratio r=12.

    Vertical distance of bouncing ball:

Relative complement or difference between sets

A\B = A-B



    Relative complement or difference between sets:

Visualizing derivatives

For a curved slope, the derivative may be  a straight line that has a gradient.
For a straight line with a gradient,  the derivative is a constant value not equal to 0.


    Visualizing derivatives exercise: Exercise available at https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/e/visualizing_derivatives

Exercise made by Stephanie Chang

    Intuitively drawing the derivative of a function: