関数 \(f(x) = 3x^3 - 5x^2 + 2x - 7\) の導関数を求め、\(x = 2\) での値を評価してください。

["Finding f(2) for the Cubic Function f(x) = 3x³ - 5x² + 2x - 7", "Understanding function evaluation is a foundational skill in algebra and calculus. In this SEO-optimized article, we’ll explore how to compute ( f(2) ) for the cubic function:", "[\nf(x) = 3x^3 - 5x^2 + 2x - 7\n]", "---", "### What is Function Evaluation?", "Function evaluation means substituting a specific value for the variable ( x ) in a given function and simplifying the expression to find the output. In this case, we substitute ( x = 2 ) into ( f(x) ).", "---", "### Step-by-Step Evaluation of ( f(2) )", "Let’s compute step-by-step:", "1. Substitute ( x = 2 ) into the function:", "[\nf(2) = 3(2)^3 - 5(2)^2 + 2(2) - 7\n]", "2. Calculate each term:", "- ( 2^3 = 8 ) → ( 3 \ imes 8 = 24 )\n- ( 2^2 = 4 ) → ( -5 \ imes 4 = -20 )\n- ( 2 \ imes 2 = 4 )\n- Constant term: ( -7 )", "3. Add all the terms together:", "[\nf(2) = 24 - 20 + 4 - 7\n]", "4. Simplify:", "[\n24 - 20 = 4 \\n4 + 4 = 8 \\n8 - 7 = 1\n]", "---", "### Final Result", "[\nf(2) = \boxed{1}\n]", "---", "### Why Evaluate Functions Example-Specific?", "Evaluating ( f(2) ) helps in various real-world applications, such as:", "- Modeling physical quantities (e.g., position, velocity) at specific time points\n- Solving optimization problems in economics and engineering\n- Understanding behavior of polynomials in higher math and data analysis", "---", "### Frequently Asked Questions", "Q: How do I evaluate a polynomial at a value?\nA: Substitute the input value for ( x ), simplify using the order of operations (PEMDAS/BODMAS), and compute.", "Q: What is the significance of finding f(2)?\nA: This point is useful for graphing, determining roots, or assessing function behavior at that specific input.", "---", "### Conclusion", "Computing ( f(2) ) for ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) is straightforward: simply substitute and simplify. With this example, students and learners can solidify their function evaluation skills—essential for mastering algebra and beyond.", "Keywords: f(x) = 3x³ - 5x² + 2x - 7 evaluated at x = 2, function evaluation, polynomial calculation, cubic function, algebra example, solving f(2)", "---", "Improve your mathematical confidence—practice evaluating functions today and unlock deeper insights into calculus and applied math."]