Use the graph of a function y = f(x) to answer the questions about the derivative: the graphs equation is y=-x^2 +3 Which of the values of x listed is the best estimate for the solution to f'(x) = 0 i. x = -1.75 ii. x = 0 iii. x = 1.75 iv. x = 4 Which of the values of x listed below is the best estimate for the solution to f'(x) = −1 i. x = -2 ii. x = -1/2 iii. x = 1/2 iv. x = 3
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.
Use the graph of a function y = f(x) to answer the questions about the derivative:
the graphs equation is y=-x^2 +3
Which of the values of x listed is the best estimate for the solution to f'(x) = 0
i. x = -1.75
ii. x = 0
iii. x = 1.75
iv. x = 4
Which of the values of x listed below is the best estimate for the solution to f'(x) = −1
i. x = -2
ii. x = -1/2
iii. x = 1/2
iv. x = 3
Result: Let .
If , then it is a upward parabola, if it is a downward parabola.
The given function is:
The given equation is a downward parabola.
Find the values of for some values of :
Plot the found points in the graph:
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Draw a parabola passing through the points:
Result: The slope of a function at is:
If the slope of a function at is , then the tangent line of at is parallel to the line .
(a) If , then slope of the function at is and the tangent line at is parallel to the line .
Notice that the tangent line of at is parallel to the line .
.png)
Hence, .
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