given
$f(x) = 4x - 4$
and $f(0) = 1$ and $f(0) = 4$.

find $f(x) =$
and find $f(3) =$

Question

given
$f(x) = 4x - 4$
and $f(0) = 1$ and $f(0) = 4$.

find $f(x) =$
and find $f(3) =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Antiderivatives", "Power Rule for Integration" ], "new_concepts": [], "current_concepts": [ "Antiderivatives", "Power Rule for Integration", "Initial Value Problems" ] } ## Find the first derivative $f'(x)$ $$ \begin{aligned} f'(x) &= \int f''(x) \, dx = \int (4x - 4) \, dx \ f'(x) &= 2x^2 - 4x + C_1 \ f'(0) &= 1 \implies 2(0)^2 - 4(0) + C_1 = 1 \implies C_1 = 1 \ f'(x) &= 2x^2 - 4x + 1 \end{aligned} $$ ## Find the function $f(x)$ $$ \begin{aligned} f(x) &= \int f'(x) \, dx = \int (2x^2 - 4x + 1) \, dx \ f(x) &= \frac{2}{3}x^3 - 2x^2 + x + C_2 \ f(0) &= 4 \implies \frac{2}{3}(0)^3 - 2(0)^2 + 0 + C_2 = 4 \implies C_2 = 4 \ f(x) &= \frac{2}{3}x^3 - 2x^2 + x + 4 \end{aligned} $$ ## Evaluate $f(3)$ $$ \begin{aligned} f(3) &= \frac{2}{3}(3)^3 - 2(3)^2 + 3 + 4 \ f(3) &= \frac{2}{3}(27) - 2(9) + 7 \ f(3) &= 18 - 18 + 7 \ f(3) &= 7 \end{aligned} $$ $f'(x) = 2x^2 - 4x + 1$ $f(3) = 7$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Antiderivatives" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the first derivative \(f'(x)\)

Find the function \(f(x)\)

Evaluate \(f(3)\)

Answer: