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The derivative of the function is defined by . If , then use a calculator to find the value of to the nearest thousandth.Answer:
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Q. The derivative of the function is defined by . If , then use a calculator to find the value of to the nearest thousandth.Answer:
- Integrate : To find , we need to integrate to get and then use the initial condition to find the constant of integration.Integrate .Integration by parts is needed here. Let and . Then and .Using integration by parts formula , we get:
- Integrate second part: Now we need to integrate the second part, . This requires another integration by parts. Let and . Then and . Applying integration by parts again, we get: .
- Integrate remaining part: Next, we integrate . This is a straightforward integration. The integral of is , so we get: .
- Find constant : Now we integrate the remaining part, . The integral of is , so we get: , where is the constant of integration.
- Calculate right side: Combining all the parts, we get the antiderivative of :$f(x) = (x^\(3\) - \(2\))(\frac{\(1\)}{\(2\)})\sin(\(2\)x) - x^\(2\)(-\frac{\(3\)}{\(4\)})\cos(\(2\)x) - (-\frac{\(3\)}{\(4\)})(x)(\frac{\(1\)}{\(2\)})\sin(\(2\)x) + (\frac{\(3\)}{\(8\)})\cos(\(2\)x) + C.
- Calculate right side: Combining all the parts, we get the antiderivative of \(f'(x)\):\[f(x) = (x^3 - 2)(\frac{1}{2})\sin(2x) - x^2(-\frac{3}{4})\cos(2x) - (-\frac{3}{4})(x)(\frac{1}{2})\sin(2x) + (\frac{3}{8})\cos(2x) + C.\]Use the initial condition \(f(6) = 6\) to find the constant \(C\). We substitute \(x = 6\) into the antiderivative and set it equal to \(6\):\[6 = (6^3 - 2)(\frac{1}{2})\sin(2\cdot6) - 6^2(-\frac{3}{4})\cos(2\cdot6) - (-\frac{3}{4})(6)(\frac{1}{2})\sin(2\cdot6) + (\frac{3}{8})\cos(2\cdot6) + C.\]Now we calculate the right side using a calculator to find \(C\).
- Calculate right side: Combining all the parts, we get the antiderivative of \(f'(x)\):\[f(x) = (x^3 - 2)(\frac{1}{2})\sin(2x) - x^2(-\frac{3}{4})\cos(2x) - (-\frac{3}{4})(x)(\frac{1}{2})\sin(2x) + (\frac{3}{8})\cos(2x) + C.\]Use the initial condition \(f(6) = 6\) to find the constant \(C\). We substitute \(x = 6\) into the antiderivative and set it equal to \(6\):\[6 = (6^3 - 2)(\frac{1}{2})\sin(2\cdot6) - 6^2(-\frac{3}{4})\cos(2\cdot6) - (-\frac{3}{4})(6)(\frac{1}{2})\sin(2\cdot6) + (\frac{3}{8})\cos(2\cdot6) + C.\]Now we calculate the right side using a calculator to find \(C\). After calculating the right side, we find the value of \(C\). Let's assume the calculation is correct and we have found \(C\). (The actual calculation would require a calculator and is not shown here.)
