If we stand on top of a 160-foot building and throw a ball up into the air with an initial velocity of 22 feet per second, then the height of the ball above the ground is given by the following. H = −16t2 + 22t + 160 Here t is time in seconds, and H is measured in feet. (a) Complete the following table giving the value H at time t. t H 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 b) Make a graph of H versus t from 0 to 4. (c) Find the value of t that maximizes H. (Round your answer to two decimal places.) t = sec (d) Find the maximum value of H. (Round your answer to two decimal places.) H = ft Explain the meaning of the number in practical terms. (e) At what times t is the height of the ball 165 feet? (Round your answers to two decimal places.)
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
If we stand on top of a 160-foot building and throw a ball up into the air with an initial velocity of 22 feet per second, then the height of the ball above the ground is given by the following.
Here t is time in seconds, and H is measured in feet.
| t | H |
|---|---|
| 0 | |
| 0.25 | |
| 0.5 | |
| 0.75 | |
| 1 | |
| 1.25 | |
| 1.5 | |
| 1.75 | |
| 2 | |
| 2.25 | |
| 2.5 | |
| 2.75 | |
| 3 | |
| 3.25 | |
| 3.5 | |
| 3.75 | |
| 4 |
b) Make a graph of H versus t from 0 to 4.
(c) Find the value of t that maximizes H. (Round your answer to two decimal places.)
t = sec
(d) Find the maximum value of H. (Round your answer to two decimal places.)
H = ft
Explain the meaning of the number in practical terms.
(e) At what times t is the height of the ball 165 feet? (Round your answers to two decimal places.)
| t | = | sec (smaller value) |
| t | = | sec (larger value) |
Step by step
Solved in 5 steps
