Physics In Exercises 67–70, (a) use the positionequation s = −16t2 + v0t + s0 to write a function thatrepresents the situation, (b) use a graphing utility tograph the function, (c) find the average rate of changeof the function from t1 to t2, (d) describe the slope of thesecant line through t1 and t2, (e) find the equation of thesecant line through t1 and t2, and (f ) graph the secant linein the same viewing window as your position function.67. An object is thrown upward from a height of 6 feet at avelocity of 64 feet per second.t1 = 0, t2 = 368. An object is thrown upward from a height of 6.5 feet ata velocity of 72 feet per second.t1 = 0, t2 = 469. An object is thrown upward from ground level at avelocity of 120 feet per second.t1 = 3, t2 = 570. An object is dropped from a height of 80 feet.t1 = 1, t2 = 2
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.
Physics In Exercises 67–70, (a) use the position
equation s = −16t2 + v0t + s0 to write a
represents the situation, (b) use a graphing utility to
graph the function, (c) find the average rate of change
of the function from t1 to t2, (d) describe the slope of the
secant line through t1 and t2, (e) find the equation of the
secant line through t1 and t2, and (f ) graph the secant line
in the same viewing window as your position function.
67. An object is thrown upward from a height of 6 feet at a
velocity of 64 feet per second.
t1 = 0, t2 = 3
68. An object is thrown upward from a height of 6.5 feet at
a velocity of 72 feet per second.
t1 = 0, t2 = 4
69. An object is thrown upward from ground level at a
velocity of 120 feet per second.
t1 = 3, t2 = 5
70. An object is dropped from a height of 80 feet.
t1 = 1, t2 = 2
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