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MTH 2205: Final Practice 3 (Part 2: Calculators are allowed)

Problem 1. \(\$1200\) is invested for \(10\) years at an interest rate of \(7\frac12\%\) compounded MONTHLY. Determine, to the nearest dollar, the total amount that accumulates.

(A) \(\$2540\) \(\quad\quad\) (B) \(\$2534\) \(\quad\quad\) (C) \(\$2473\) \(\quad\quad\) (D) \(\$2408\) \(\quad\quad\) (E) \(\$2280\)

Problem 2. Evaluate \(\lim_{x\to 0}\frac{x^2}{\left(e^x-1\right)^2}\).

(A) \(1\) \(\quad\quad\) (B) \(0\) \(\quad\quad\) (C) \(\infty\) \(\quad\quad\) (D) undefined \(\quad\quad\) (E) e

Problem 3. Find the critical numbers of \(f(x)=\frac{e^x-x^6}{10000}\).

(A) \(0.482\) only \(\quad\quad\) (B) \(0.482\) and \(13.94\) \(\quad\quad\) (C) \(1\) \(\quad\quad\) (D) \(0.824\) only \(\quad\quad\) (E) \(0.824\) and \(15.494\)

Problem 4. The graphs shown below display different graphical windows. More that one may actually be the graphs of the function \(f\) from the previous problem. Pick the graph that most accurately portrays the function (shows all its relative extrema, asymptotes, etc.).

(A) \(\quad\quad\) (B) \(\quad\quad\)
(C) \(\quad\quad\) (D) \(\quad\quad\)

Problem 5. To three decimal places, \(\log_312=\)

(A) \(1.573\) \(\quad\quad\) (B) \(2.262\) \(\quad\quad\) (C) \(3.145\) \(\quad\quad\) (D) \(4.879\) \(\quad\quad\) (E) \(5.741\)

Problem 6. If \(f(x)=\frac{12\ln\left(3x^2+2e^{4x^2+7}\right)}{\sqrt{3x^2+1}}\), then to two decimal places, \(f'(1)=\)

(A) \(-3.67\) \(\quad\quad\) (B) \(2.69\) \(\quad\quad\) (C) \(12.86\) \(\quad\quad\) (D) \(-4.62\) \(\quad\quad\) (E) none of these

Problem 7. If \(x\) dollars are spent on advertising, then the revenue in dollars is given by \(R=\frac{100000}{1+2^{15-0.005x}}\), where \(0\leq x\leq 5000\). How much is spent on advertising at the point of diminishing returns (point of inflection)?

(A) \(\$10576\) \(\quad\quad\) (B) \(\$100000\) \(\quad\quad\) (C) \(\$50086.6\) \(\quad\quad\) (D) \(\$3000\) \(\quad\quad\) (E) There is no point of diminishing returns

Problem 8. The area of the region bounded by \(f(x)=x^2\), the \(x\)-axis and the lines \(x=1\) and \(x=5\) is approximated by \(n=4\) rectangles. If the right endpoint is used for each of these rectangles, then the approximate area obtained is

(A) \(30\) \(\quad\quad\) (B) \(41\) \(\quad\quad\) (C) \(42\) \(\quad\quad\) (D) \(54\) \(\quad\quad\) (E) \(64\)

Problem 9. Given \(g(x)=\frac{x^5+3x^3+4x^2+1}{x^4-x^2+2}\), find \(g''(0.5)\).

(A) \(3.98692\) \(\quad\quad\) (B) \(1297.5\) \(\quad\quad\) (C) \(-122.438\) \(\quad\quad\) (D) \(0\) \(\quad\quad\) (E) \(12.2258\)

Problem 10. Linearize \(f(x)=\left(x^3+2x+1\right)^{1.7}\) near \(x=0\).

(A) \(3.4x+1\) \(\quad\quad\) (B) \(1.7x+1\) \(\quad\quad\) (C) \(2x+1\) \(\quad\quad\) (D) \(1.7x\) \(\quad\quad\) (E) \(1\)