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Chapter 4: Problem 68

Find the solution of the following initial value problems. $$g^{\prime}(x)=7 x^{6}-4 x^{3}+12 ; g(1)=24$$

Short Answer

Expert verified
Question: Determine the function g(x) given its first derivative is g'(x) = 7x^6 - 4x^3 + 12 and an initial value of g(1) = 24. Answer: The function g(x) is given by g(x) = x^7 - x^4 + 12x + 12.

Step by step solution

01

Integrate g'(x)

Integrate the given function with respect to x: $$\int (7x^6 - 4x^3 + 12) dx = \int 7x^6 dx - \int 4x^3 dx + \int 12 dx$$ Now, integrate each term individually: $$7\int x^6 dx - 4\int x^3 dx + \int 12 dx$$ $$7\frac{x^7}{7} - 4\frac{x^4}{4} + 12x + C$$ After simplifying, we get: $$g(x) = x^7 - x^4 + 12x + C$$
02

Use the initial value to determine the constant of integration

We know that g(1) = 24. Let's plug in this initial value into the function: $$24 = g(1) = (1)^7 - (1)^4 + 12(1) + C$$ Now, we solve for C: $$24 = 1 - 1 + 12 + C$$ $$C = 24 - 12 = 12$$
03

Write down the final solution, g(x)

Now that we have the constant of integration, let's plug it into the function to find the final solution: $$g(x) = x^7 - x^4 + 12x + 12$$

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