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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: Find the function g(x) given that g'(x) = 7x^6 - 4x^3 + 12 and g(1) = 24. Answer: g(x) = x^7 - x^4 + 12x + 12

Step by step solution

01

Integrate g'(x) to find g(x)

First, we need to integrate g'(x) to find g(x). We are given: $$g'(x) = 7x^6 - 4x^3 + 12$$ Integration of each term gives us : $$g(x) = \int (7x^6 - 4x^3 + 12) dx = \frac{7x^7}{7} - \frac{4x^4}{4} + 12x + C$$ Simplify the expression: $$g(x) = x^7 - x^4 + 12x + C$$
02

Apply the initial condition g(1) = 24

Now we'll use the initial condition, g(1) = 24, to solve for the constant C. Plug in x = 1 into the expression for g(x): $$24 = (1)^7 - (1)^4 + 12(1) + C$$
03

Solve for C

The equation from the previous step becomes: $$24 = 1 - 1 + 12 + C$$ Simplify and solve for C: $$C = 24 - 12 = 12$$
04

Write the final solution

Now that we have the value of the constant C, we can write the final solution for g(x): $$g(x) = x^7 - x^4 + 12x + 12$$

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