Example Question - polynomial function
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Determining the Derivative of a Polynomial Function
<p>The derivative of \(y = 2t^4 - 10t^2 + 13t\) with respect to \(t\) is calculated by differentiating each term separately.</p> <p>The derivative of the first term \(2t^4\) is \(8t^3\).</p> <p>The derivative of the second term \(-10t^2\) is \(-20t\).</p> <p>The derivative of the third term \(13t\) is \(13\).</p> <p>Therefore, the derivative \( \frac{dy}{dt} = 8t^3 - 20t + 13 \).</p>
Derivative of a Polynomial Function
Given: y = 2t^4 - 10t^2 + 13t. To find: dy/dt. Solution: dy/dt = d/dt (2t^4) - d/dt (10t^2) + d/dt (13t) dy/dt = 2 * 4t^(4-1) - 10 * 2t^(2-1) + 13 * 1t^(1-1) dy/dt = 8t^3 - 20t + 13.
Derivative of a Polynomial
\[ \frac{dy}{dt} = \frac{d}{dt}(2t^4) - \frac{d}{dt}(10t^2) + \frac{d}{dt}(13t) \\ \frac{dy}{dt} = 8t^3 - 20t + 13 \]
Derivative of a polynomial function
\frac{dy}{dt} = \frac{d}{dt}(2t^4) - \frac{d}{dt}(10t^2) + \frac{d}{dt}(13t) \frac{dy}{dt} = 8t^3 - 20t + 13
Solving for Constants in a Polynomial Function
The image contains a question that asks you to find the values of constants 'a' and 'b' in a polynomial function \( P(x) = 3x^3 + ax^2 + bx - 6 \), given that \( x - 2 \) is a factor of \( P(x) \) and that the remainder is 27 when \( P(x) \) is divided by \( (x + 2) \). To solve this: 1. If \( x - 2 \) is a factor of \( P(x) \), it means that \( P(2) = 0 \). 2. If the remainder is 27 when \( P(x) \) is divided by \( x + 2 \), it means that \( P(-2) = 27 \). Let's use these conditions to form two equations to solve for 'a' and 'b'. 1. \( P(2) = 0 \) \[ 3(2)^3 + a(2)^2 + b(2) - 6 = 0 \] \[ 3(8) + 4a + 2b - 6 = 0 \] \[ 24 + 4a + 2b = 6 \] \[ 4a + 2b = 6 - 24 \] \[ 4a + 2b = -18 \] \[ 2a + b = -9 \] (Equation 1) 2. \( P(-2) = 27 \) \[ 3(-2)^3 + a(-2)^2 + b(-2) - 6 = 27 \] \[ 3(-8) + 4a - 2b - 6 = 27 \] \[ -24 + 4a - 2b = 27 + 6 \] \[ 4a - 2b = 33 + 24 \] \[ 4a - 2b = 57 \] \[ 2a - b = 57/2 \] (Equation 2) Now we have a system of linear equations: \[ 2a + b = -9 \] (Equation 1) \[ 2a - b = 57/2 \] (Equation 2) Multiply Equation 1 by 2 and add it to Equation 2: \[ 4a + 2b = -18 \] (Equation 1 transformed) \[ 2a - b = 57/2 \] (Equation 2) \[ (4a + 2b) + (2a - b) = -18 + 57/2 \] \[ 6a + b = 21/2 \] (Equation 3) Now subtract Equation 1 from Equation 3: \[ 6a + b - (2a + b) = 21/2 + 9 \] \[ 4a = 21/2 + 18/2 \] \[ 4a = 39/2 \] \[ a = \frac{39/2}{4} \] \[ a = 39/8 \] \[ a = 4.875 \] Now we can substitute the value of 'a' back into Equation 1 to solve for 'b': \[ 2a + b = -9 \] \[ 2(4.875) + b = -9 \] \[ 9.75 + b = -9 \] \[ b = -9 - 9.75 \] \[ b = -18.75 \] The values are \( a \approx 4.875 \) (or \( a = 39/8 \)) and \( b \approx -18.75 \) (or \( b = -75/4 \)).
Solving Integral of Polynomial Function
The image shows an integral that needs to be solved. The expression within the integral is a polynomial function: ∫(8x^3 - x^2 + 5x - 1) dx To solve this, we integrate each term of the polynomial separately with respect to x. The integral of a constant times a power of x is the constant times the fraction of the incremented power of x divided by the new exponent. The integral of a constant is just the constant times x. Here's how this integral is computed: ∫8x^3 dx = 2x^4 (because you add 1 to the exponent and then divide by the new exponent) ∫-x^2 dx = -x^3/3 (same principle) ∫5x dx = 5x^2/2 (again, same principle) ∫-1 dx = -x Putting it all together, the antiderivative of the given function is: 2x^4 - x^3/3 + 5x^2/2 - x + C where C is the constant of integration.
Solving Indefinite Integral of a Polynomial Function
The question in the image is asking for the integral of the given function with respect to x: ∫(8x^3 - x^2 + 5x - 1) dx To solve this indefinite integral, you would integrate each term separately. The integral of a sum is the sum of the integrals, so you can integrate each term individually. The integral of 8x^3 with respect to x is (8/4)x^(3+1) = 2x^4. The integral of -x^2 with respect to x is -(1/3)x^(2+1) = -1/3 x^3. The integral of 5x with respect to x is (5/2)x^(1+1) = 5/2 x^2. The integral of -1 with respect to x is -x. Putting it all together, the indefinite integral of the function is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C where C is the constant of integration.
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