Mathematics

Mathematics

Problem

Let gg be a differentiable function for which dgdx=11+x2+2x\dfrac{dg}{dx}=\dfrac{1}{1+x^{2}}+2x and g(0)=−1g(0)=-1. (a) Determine the explicit expression for g(x)g(x) that satisfies the given condition. (b) Compute g(1)g(1), showing all steps in your work. (c) Describe the end behavior of g(x)g(x) as x→∞x\to\infty, supporting your conclusion algebraically.
Intuitive explanation

The key is recognizing that gg is defined by its derivative and an initial condition—a classic setup for finding an antiderivative. Notice that dgdx\frac{dg}{dx} is a sum of two simple pieces: 11+x2\frac{1}{1+x^2} (which has a standard antiderivative) and 2x2x (a basic power function). Integrate both terms separately, then use g(0)=−1g(0)=-1 to pin down the constant of integration.

Intuitive explanation

The key is recognizing that gg is defined by its derivative and an initial condition—a classic setup for finding an antiderivative. Notice that dgdx\frac{dg}{dx} is a sum of two simple pieces: 11+x2\frac{1}{1+x^2} (which has a standard antiderivative) and 2x2x (a basic power function). Integrate both terms separately, then use g(0)=−1g(0)=-1 to pin down the constant of integration.

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