for which interval is the average rate of change of $f(x)$ negative?

from $x = -3.5$ to $x = -1$
from $x = -3$ to $x = 3$
from $x = 0$ to $x = 2.5$

Question

for which interval is the average rate of change of $f(x)$ negative?

from $x = -3.5$ to $x = -1$
from $x = -3$ to $x = 3$
from $x = 0$ to $x = 2.5$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Average Rate of Change" ], "new_concepts": [], "current_concepts": [ "Average Rate of Change", "Function Evaluation", "Polynomial Functions" ] } ## Define average rate of change Using the *Average Rate of Change* knowledge point $$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$ ## Evaluate the first interval Using the *Average Rate of Change* knowledge point $$ \begin{aligned} &\text{Interval: } [a, b] = [-3.5, -1]\ &f(-3.5) \approx -2.5\ &f(-1) \approx 0\ &\frac{f(-1) - f(-3.5)}{-1 - (-3.5)} \approx \frac{0 - (-2.5)}{2.5} = 1 > 0 \end{aligned} $$ ## Evaluate the second interval Using the *Average Rate of Change* knowledge point $$ \begin{aligned} &\text{Interval: } [a, b] = [-3, 3]\ &f(-3) = -3\ &f(3) = -3\ &\frac{f(3) - f(-3)}{3 - (-3)} = \frac{-3 - (-3)}{6} = 0 \end{aligned} $$ ## Evaluate the third interval Using the *Average Rate of Change* knowledge point $$ \begin{aligned} &\text{Interval: } [a, b] = [0, 2.5]\ &f(0) = 1\ &f(2.5) \approx -2.5\ &\frac{f(2.5) - f(0)}{2.5 - 0} \approx \frac{-2.5 - 1}{2.5} = -1.4 < 0 \end{aligned} $$

from $x = -3.5$ to $x = -1$

from $x = -3$ to $x = 3$

from $x = 0$ to $x = 2.5$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Average Rate of Change" ] }

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QUESTION IMAGE

Question

Explanation:

Define average rate of change

Evaluate the first interval

Evaluate the second interval

Evaluate the third interval

Answer: