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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 for : To find the value of , we need to integrate the derivative to get the original function . We will then use the initial condition to find the constant of integration.
- Use Symbolic Integration: The integral of is a bit complex and typically requires integration by parts or a special technique. However, since we are not given the exact method for integration and are asked to use a calculator, we will assume that we can use a calculator that can handle symbolic integration.
- Apply Initial Condition: Using a calculator with symbolic integration capability, we integrate to find . The antiderivative of is not straightforward, but we can write it as:, where is the constant of integration.
- Calculate Definite Integral: We apply the initial condition to solve for . We substitute into the integrated function and set it equal to :-6 = f(6) = \int_0^6 x^2\cos(3x) \, dx + C\.We calculate the definite integral from \$0 to and then solve for .
- Find Constant : After calculating the definite integral from to , we find the value of by solving the equation for . Let's assume the calculator gives us the value of the integral, and we find accordingly.
- Evaluate : Now that we have the constant , we can find by evaluating the antiderivative at and adding the constant :
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We calculate the definite integral from to using the calculator and then add the constant to find . - Round to Nearest Thousandth: We round the result to the nearest thousandth as requested.
