Change

Exercise 1

You are given the following graph of a function.
Determine

over which intervals the function is increasing or decreasing, and which type of increase (decrease) you observe;
the approximate values of any extrema of the function;
at what values of $x$ you observe the greatest rate of increase or decrease.

Exercise 2

You are given the function rule $f (x) = 18 x - x^{3}$ .
Determine

over which intervals the function is increasing or decreasing, and which type of increase (decrease) you observe;
the values of any extrema of the function;
at what values of $x$ you observe the greatest rate of increase or decrease.

Exercise 3

You are given a function $f$ with rule $f (x) = 0, 5 x^{4} - 4 x^{2} + 8$ .

a

You can use your graphing calculator to look at the curve for this function. What are the extrema of the function?

b

The function has exactly one interval with an accelerated decrease. Which interval is this?

c

You can often deduce the range of a function if you know the values of its extrema. What is the (probable) range of the above function?

Exercise 4

Here you see a graph of a parachute jump from a height of $3500$ meter. After a period of free fall the jumper opened his parachute.

a

After how many seconds did the jumper open his parachute? How do you know this from looking at the graph?

b

During free fall, the curve shows an accelerated decrease. What does that tell you about the speed of falling in that interval?

c

After the parachute has opened, the jumper continues to fall with a constant speed. How do you know this from looking at the graph? What is the speed in that interval?

Exercise 5

You have the following information about the temperature $T$ in °C on a given day:

at 6 a.m. ( $t = 6$ ) the temperature was $T = 2$ °C;
the graph shows an accelerated increase from $t = 6$ until $t = 12$ ;
the graph shows a decelerated increase from noon until 2.30 p.m. ( $t = 14,5$ ), and then moves to an accelerated decrease until $t = 20$ ;
the graph shows a decelerated decrease from $t = 20$ until the end of the day.

Make a sketch of what the graph corresponding to this function might look like, and then explain at what value of $t$ the function $T$ should have an extremum.

Exercises