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:

Range of a function

Use differentiation, to find the vertex.

plot2d in Maxima can also be used to find the shape of the graph which can help to find the range.

    Domain and range of a function:

Extreme value theorem

The absolute maximum or minimum value could also be at the beginning or end of the function.

 plot2d in Maxima  can be used to solve some of the problems for this topic.

    Extreme value theorem:

How many ways can the letters in the word mathematics be arranged?

11!/(2!2!2!)=11!/8=4989600

 Divide by two factorial for each of the three letters which appear twice: m,a,t.