Example Question - polynomial functions
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Understanding Quadratic Equations
An equation that can be written in the form \( ax^2 + bx + c = 0 \) (with \( a \neq 0 \)) is called a quadratic equation.
Behavior of Exponential and Polynomial Functions in Limits
The limit presented in the image is: \[ \lim_{{x \to \infty}} \frac{3e^x - 2}{x^3} \] To solve this, we need to assess the behavior of the numerator and the denominator as x approaches infinity. The term \(3e^x\) grows exponentially as x approaches infinity, while the \(x^3\) term grows at a cubic rate. Since exponential growth (in this case \(e^x\)) outpaces any polynomial growth (\(x^3\)), the numerator will become infinitely larger compared to the denominator. Therefore, as \(x\) approaches infinity, the fraction \(\frac{3e^x - 2}{x^3}\) will approach infinity. Thus, the limit is: \[ \lim_{{x \to \infty}} \frac{3e^x - 2}{x^3} = \infty \]
Definite Integrals of Polynomial Functions
The image shows an integral which needs to be solved. The function to be integrated is a polynomial: 8x^3 - x^2 + 5x - 1. To solve the integral, we need to integrate each term individually with respect to x. Here's how you integrate each term: 1. ∫8x^3 dx = (8/4)x^4 = 2x^4 2. ∫(-x^2) dx = -(1/3)x^3 3. ∫5x dx = (5/2)x^2 4. ∫(-1) dx = -x Now, putting it all together: ∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C represents the constant of integration.
Evaluating Definite Integrals of Polynomial Functions using Power Rule
This is a definite integral of a polynomial function, which we can evaluate using the Power Rule for integration. The Power Rule states that the integral of x^n is (x^(n+1)) / (n+1) plus a constant (C), for any real number n ≠ -1. So let's evaluate your integral: ∫(8x^3 - x^2 + 5x - 1) dx Integration term by term: ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) = 2 * x^4 ∫-x^2 dx = - ∫x^2 dx = - (x^(2+1) / (2+1)) = - (1/3) * x^3 ∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) = (5/2) * x^2 ∫-1 dx = -x Combining these results: 2x^4 - (1/3)x^3 + (5/2)x^2 - x Without limits of integration given, we cannot evaluate for specific numbers. If there were limits, you would substitute the upper limit into the result for x, then subtract the result from substituting the lower limit. Since no limits are provided, this is the indefinite integral result, and don't forget to add the constant of integration C: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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