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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) $$\quad\quad$$
(E) 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) $$1$$