Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Answer:
7
Step-by-step explanation:
f(x) = 2x^3 + x^2 − 3x + 1
f'(x) = 6x^2 + 2x − 3
The slope at -2:
f'(-2) = 6(-2)^2 + 2(-2) − 3
f'(-2) = 24 + -4 − 3
f'(-2) = 17
-----
The slope at 0:
f'(0) = 6(0)^2 + 2(0) − 3
f'(0) = − 3
The average rate of change is (17+(-3))/2, or 14/2, or 7
Step-by-step explanation:
In calculus, the derivative corresponds to the slope of the tangent line at a point on a function of which also indicates the concept of an instantaneous rate of change. However, by its definition, a tangent line is a line that intersects the function at only one point at which its slope would be impossible to be determined analytically using analytical geometry.
In analytic geometry, the slope of a line passing through two given points [tex](x_{1}, \ f(x_{1}))[/tex] and [tex](x_{2}, \ f(x_{2}))[/tex] on the function [tex]f(x)[/tex] , also known as a secant line, is evaluated by the formula
[tex]\text{slope} \ = \ \displaystyle\frac{f(x_{2}) \ - \ f(x_{1})}{x_{2} \ - \ x_{1}}[/tex].
In the case of the slope of a tangent line, [tex]x_{1} \ = \ x_{2}[/tex] and therefore [tex]f(x_{1}) \ = \ f(x_{2})[/tex]. Plugging this consensus into the formula above, yields
[tex]\text{slope} \ = \ \displaystyle\frac{0}{0}[/tex],
which is an indeterminate form where the slope is undefined.
Therefore, we can only approximate the slope of a tangent line using the secant line and evaluate the corresponding slope when one of the two points on the function moves closer and closer towards the other point but will not be equal to each other.
Hence, contrary to the tangent line, a secant line shows the average rate of change between two distinct points on the provided function.
Therefore, the average rate of change from the point x = -2 to the point x = 0 is
[tex]rate_{avg} \ = \ \displaystyle\frac{\left[2(0)^{3} \ + \ (0)^{2} \ - \ 3(0) \ + \ 1\right] \ - \ \left[2(-2)^{3} \ + \ (-2)^{2} \ - \ 3(-2) \ + \ 1\right]}{0 \ - \ (-2)} \\ \\ rate_{avg} \ = \ \displaystyle\frac{1 \ - \ (-5)}{2} \\ \\ rate_{avg} \ = \ \displaystyle\frac{6}{2} \\ \\ rate_{avg} \ = \ 3[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.