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كويز تفاعلي: Polynomial Functions Properties and Extrema ريفيل
This test evaluates knowledge of polynomial functions, including identifying degrees and leading coefficients, and locating relative maxima and minima. Designed based on materials by Mr. Karam Asaad.
يرجى الانتباه إلى أن المعلم قام بإعداد الأسئلة فقط، ولم يقم بإعداد الإجابات أو الشروحات المرفقة. وقد تم توليد الإجابات باستخدام تقنيات الذكاء الاصطناعي، لذلك قد تتضمن بعض الأخطاء أو عدم الدقة.
للحصول على الإجابات الصحيحة والمضمونة، يُرجى الرجوع إلى المعلم أو المصدر الدراسي المعتمد.
Question 1
Points: 1
Describe end behavior, degree, and leading coefficient. For f(x) = -5x4 + 3x2, what is the degree?
Explanation
The degree of a polynomial is the highest power of the variable present in the function. In f(x) = -5x4 + 3x2, the highest exponent is 4.
Question 2
Points: 1
For g(x) = 2x5 + 6x4, what is the leading coefficient?
Explanation
The leading coefficient is the coefficient of the term with the highest degree. For g(x) = 2x5 + 6x4, the term with the highest degree is 2x5, so the leading coefficient is 2.
Question 3
Points: 1
For g(x) = 8x4 + 5x5, what is the degree?
Explanation
The degree is the highest exponent in the polynomial. In g(x) = 8x4 + 5x5, the term with the highest exponent is 5x5, so the degree is 5.
Question 4
Points: 1
For h(x) = 9x6 - 5x7 + 3x2, what is the leading coefficient?
Explanation
Identify the term with the highest degree, which is -5x7. The coefficient of this term is -5.
Question 5
Points: 1
Use a table to graph. Estimate x-coordinates of relative maxima and minima. For f(x) = -2x3 + 12x2 - 8x, where does a relative minimum occur (approx.)?
Explanation
Calculating the derivative $f'(x) = -6x^2 + 24x - 8$ and solving for roots gives $x \approx 0.37$ and $x \approx 3.63$. Using the second derivative test, $f''(0.37) > 0$, confirming a relative minimum near x = 0.4.
Question 6
Points: 1
For f(x) = -2x3 + 12x2 - 8x, where does a relative maximum occur (approx.)?
Explanation
The critical points are at $x \approx 0.37$ and $x \approx 3.63$. Since $f''(3.63) < 0$, a relative maximum occurs there, which is approximately x = 4 among the provided choices.
Question 7
Points: 1
For f(x) = x4 + 2x - 1, the graph has a relative minimum near:
Explanation
The derivative is $f'(x) = 4x^3 + 2$. Setting $f'(x) = 0$ results in x3 = -0.5, which gives $x \approx -0.79$, rounding to -0.8.
Question 8
Points: 1
For f(x) = x4 + 8x2 - 12, is there a relative minimum?
Explanation
The derivative $f'(x) = 4x^3 + 16x = 4x(x^2 + 4)$ has only one real root at x = 0. Since $f''(0) = 16 > 0$, the point at x = 0 is a relative minimum.
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