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Chapter 2: Problem 2

For the function \(\mathrm{f}(\mathrm{x})=1+3^{\mathrm{x}}\) /n 3 find the antiderivative \(\mathrm{F}(\mathrm{x})\), which assumes the value 7 for \(x=2\). At what values of \(x\) does the curve \(F(x)\) cut the \(x\)-axis?

Short Answer

Expert verified
Question: Find the x-values where the antiderivate \(F(x)\) of the function \(f(x) = 1 + 3^x\) intersects the x-axis, given \(F(2) = 7\). Answer: The x-values where the antiderivative \(F(x)\) intersects the x-axis cannot be determined using a closed-form solution. Use a numerical method or graphical approach to find an approximate value for x.

Step by step solution

01

Find the General Form of the Antiderivative of f(x)

To find the general form of the antiderivative of \(f(x)\), we can find the integral of \(f(x)\) with respect to \(x\). The general form of the antiderivative of \(f(x) = 1 + 3^x\) is given by: \(F(x) = \int (1 + 3^x) dx = \int1 dx + \int3^x dx\) Now, we can find the integral of each term. The integral of 1 with respect to x is simply x. To find the integral of \(3^x\) with respect to x, we can apply the formula: \(\int a^x dx = \frac{a^x}{\ln a} + C\) Thus, \(F(x) = x + \frac{3^x}{\ln 3} + C\)
02

Find the Particular Form of the Antiderivative using the Given Point

We are given that \(F(2) = 7\). We can use this information to find the value of the constant C: \(7 = 2 + \frac{3^2}{\ln 3} + C\) Now, we can solve for C: \(C = 7 - 2 - \frac{3^2}{\ln 3} = 5 - \frac{9}{\ln 3}\) So, the particular form of the antiderivative is: \(F(x) = x + \frac{3^x}{\ln 3} + 5 - \frac{9}{\ln3}\)
03

Find the Values of x where F(x) Intersects the x-axis

To find the values of x where \(F(x)\) intersects the x-axis, we set \(F(x)\) equal to 0 and solve for x: \(0 = x + \frac{3^x}{\ln 3} + 5 - \frac{9}{\ln3}\) Subtracting 5 from both sides, and multiplying both sides by \(\ln 3\), we get: \(-5\ln 3 = x\ln 3 + 3^x - 9\) Unfortunately, this equation does not have a straightforward closed-form solution. We have to use a numerical method like the Newton-Raphson method or a graphical approach to find an approximate value for x where the curve \(F(x)\) intersects the x-axis.

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