Question

Use the following graph of the function f(x) = 2x3 + x2 − 3x + 1 to answer this question: graph of 2x cubed plus x squared minus 3x plus 1 What is the average rate of change from x = −2 to x = 0?

Answer 1

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

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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

                           

Answer 2

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 $(x_{1}, \ f(x_{1}))$ and $(x_{2}, \ f(x_{2}))$ on the function $f(x)$ , also known as a secant line, is evaluated by the formula

                                          $\text{slope} \ = \ \displaystyle\frac{f(x_{2}) \ - \ f(x_{1})}{x_{2} \ - \ x_{1}}$.

In the case of the slope of a tangent line, $x_{1} \ = \ x_{2}$ and therefore $f(x_{1}) \ = \ f(x_{2})$. Plugging this consensus into the formula above, yields

                                                      $\text{slope} \ = \ \displaystyle\frac{0}{0}$,

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

    $rate_{avg} \ = \ \displaystyle\frac{\left[2(0)^{3} \ + \ (0)^{2} \ - \ 3(0) \ + \ 1\right] \ - \ \left[2(-2)^{3} \ + \ (-2)^{2} \ - \ 3(-2) \ + \ 1\right]}{0 \ - \ (-2)} \\ \\ rate_{avg} \ = \ \displaystyle\frac{1 \ - \ (-5)}{2} \\ \\ rate_{avg} \ = \ \displaystyle\frac{6}{2} \\ \\ rate_{avg} \ = \ 3$.