Calculus I

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Let \(f(x) = e^x - 5x\).

1. Graph \(f(x)\).
2. Use Newton's method to approximate the smallest zero of \(f(x)\). Start with initial guess \(x_0 = 0\), and continue until two consecutive iterations are the same as far as Desmos can displays them.

Use \(\sin x\) to approximate \(\pi\) using Newton's method with initial guess 3. Continue until your answer is the same as the approximation of \(\pi\) displayed by Desmos.

Let \(g(x)= x^2\), and \(F = (3,0)\). Find the point \(P = (x,g(x))\) so that \(d(P,F)\) is minimal. What is the minimal distance? Explain how you found \(P\). Hint: the distance is minimized when the square of the distance is minimized.

Let \(h(x) = (x+2)e^{x^2}\). Find the point \(Q=(x,h(x))\) so that \(d(Q,(0,0))\) is minimal. What is the minimal distance? Hint: This is similar to Problem 3, but the zero you are looking for is not nice. Hence you may need a way to approximate the zero.