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## MTH 2205: Final Practice 3 (Part 1: No calculators)

Problem 1. The graph of $$y=f(x)$$ appears in the picture below. Estimate the points at which the absolute maximum and the absolute minimum occur on the interval $$[0,3]$$. (A) absolute minimum: $$(0,0)$$; absolute maximum $$(3,2)$$
(B) absolute minimum: $$(0,0)$$; absolute maximum $$(2,4)$$
(C) absolute minimum: $$(-2,-4)$$; absolute maximum $$(2,4)$$
(D) absolute minimum: $$(4,-5)$$; absolute maximum $$(-4,5)$$
(E) absolute minimum: $$(3,2)$$; absolute maximum $$(2,4)$$

Problem 2. Find the open interval(s) on which $$f(x)=x^3-3x+7$$ is increasing

(A) $$(-\infty,-1)$$ or $$(1,+\infty)$$ (i.e. $$x < -1$$ or $$x > 1$$) $$\quad\quad$$ (B) $$(1,+\infty)$$ only (i.e. $$x > 1$$ only) $$\quad\quad$$
(C) $$(-1,1)$$ only (i.e. $$-1 < x < 1$$) $$\quad\quad$$ (D) $$(-\infty,-1)$$ only $$\quad\quad$$ (i. e. $$x < -1$$ only) (E) $$(-\infty,+\infty)$$ (i.e. all real $$x$$)

Problem 3. The position of an object at any time $$t$$ is given by $$s(t)=-8t^2+20t+10$$. Find the acceleration when $$t=2$$.

(A) $$18$$ $$\quad\quad$$ (B) $$-12$$ $$\quad\quad$$ (C) $$-16$$ $$\quad\quad$$ (D) $$22$$ $$\quad\quad$$ (E) $$\frac 54$$

Problem 4. The position of an object at any time $$t$$ is given by $$s(t)=-8t^2+20t+10$$. Find the time when the velocity is $$0$$.

(A) $$18$$ $$\quad\quad$$ (B) $$-12$$ $$\quad\quad$$ (C) $$-16$$ $$\quad\quad$$ (D) $$22$$ $$\quad\quad$$ (E) $$\frac 54$$

Problem 5. Find the maximum profit for the profit function $$P(x)=-2x^2+10x-3$$

(A) $$10$$ $$\quad\quad$$ (B) $$\frac{19}2$$ $$\quad\quad$$ (C) $$\frac{5+\sqrt{19}}2$$ $$\quad\quad$$ (D) $$\frac 74$$ $$\quad\quad$$ (E) $$\frac{67}8$$

Problem 6. A square is measured and each side is found to be $$5$$ inches with a possible error of at most $$0.03$$ inches. Use differentials to find the approximate error in computing the area of the square

(A) $$0.6$$ $$\quad\quad$$ (B) $$0.06$$ $$\quad\quad$$ (C) $$0.006$$ $$\quad\quad$$ (D) $$0.3$$ $$\quad\quad$$ (E) $$0.03$$

Problem 7. Find the derivative of $$f(x)=3e^{-5x+7}$$

(A) $$-15e^{-5}$$ $$\quad\quad$$ (B) $$-15e^{-5x+7}$$ $$\quad\quad$$ (C) $$3e^{-5}$$ $$\quad\quad$$ (D) $$-15e^{-5x+7}+3e^{-5x+7}$$ $$\quad\quad$$ (E) $$3e^{-5}+e^{-5x+7}$$

Problem 8. Evaluate $$e^{3\ln 5}$$.

(A) $$15$$ $$\quad\quad$$ (B) $$12e$$ $$\quad\quad$$ (C) $$125$$ $$\quad\quad$$ (D) $$\frac35$$ $$\quad\quad$$ (E) $$243$$

Problem 9. Find the derivative of $$f(x)=\ln\left(\frac{x^2-7}{x}\right)$$.

(A) $$\frac1{x^2-7}-\frac1x=\frac{-x^2+x+7}{x\left(x^2-7\right)}$$ $$\quad\quad$$ (B) $$\frac1{x^2-7}+\frac1x=\frac{x^2+x-7}{x\left(x^2-7\right)}$$ $$\quad\quad$$ (C) $$\frac{2x}{x^2-7}+\frac1x=\frac{3x^2-7}{x\left(x^2-7\right)}$$ $$\quad\quad$$
(D) $$\frac{2x}{x^2-7}-\frac1x=\frac{x^2+7}{x\left(x^2-7\right)}$$ $$\quad\quad$$ (E) $$\frac x{x^2-7} + \frac1x=\frac{2x^2-7}{x\left(x^2-7\right)}$$

Problem 10. Solve $$y'=5x^3y^2$$

(A) $$y=\frac{-4}{5x^4+4C}$$ $$\quad\quad$$ (B) $$y=\frac{4}{5x^4+4C}$$ $$\quad\quad$$ (C) $$y=\frac{5x^4y^3}{12}+C$$ $$\quad\quad$$
(D) $$-\frac{5}{4}x^4+C$$ $$\quad\quad$$ (E) $$\frac54x^4+C$$

Problem 11. Evaluate $$\int_4^8\frac1{\sqrt{2t-4}}\,dt$$

(A) $$4$$ $$\quad\quad$$ (B) $$2$$ $$\quad\quad$$ (C) $$\frac1{\sqrt{10}}-\frac12$$ $$\quad\quad$$ (D) $$2\sqrt 3-2$$ $$\quad\quad$$ (E) $$2\sqrt 3+2$$

Problem 12. Find all critical points $$(x,y)$$ of the function $$f(x)=4x-x^2-2$$

(A) $$(2,2)$$ $$\quad\quad$$ (B) $$(2,-2)$$ $$\quad\quad$$ (C) $$(2,0)$$ $$\quad\quad$$ (D) $$(-2,-14)$$ $$\quad\quad$$ (E) $$(0,-2)$$

Problem 13. The function $$f$$ is the same from the previous problem, i.e. $$f(x)=4x-x^2-2$$. Find the absolute extrema on the closed interval $$[0,5]$$ (i.e. $$0\leq x\leq 5$$).

(A) minimum is $$-2$$ and maximum is $$2$$ $$\quad\quad$$ (B) minimum is $$-7$$ and maximum is $$-2$$ $$\quad\quad$$ (C) minimum is $$-7$$ and maximum is $$2$$ $$\quad\quad$$ (D) minimum is $$-7$$ and maximum is $$5$$ $$\quad\quad$$ (E) minimum is $$-10$$ and maximum is $$10$$

Problem 14. The demand function and cost function for $$x$$ units of a product are $$p=\frac{60}{\sqrt x}$$ and $$C=0.65x+400$$. Find the marginal profit when $$x=100$$.

(A) $$\2.35$$ per unit $$\quad\quad$$ (B) $$\4.58$$ per unit $$\quad\quad$$ (C) $$\193.50$$ per unit $$\quad\quad$$ (D) $$\187.35$$ per unit $$\quad\quad$$ (E) $$\3.65$$ per unit

Problem 15. If $$f(12)=200$$ and $$f'(12)=-6$$, estimate the value of $$f(14)$$ for the function $$y=f(x)$$

(A) $$194$$ $$\quad\quad$$ (B) $$188$$ $$\quad\quad$$ (C) $$206$$ $$\quad\quad$$ (D) $$214$$ $$\quad\quad$$ (E) $$8$$ and $$x=-1$$

Problem 16. Given the function $$f(x)=5e^x$$, which of the following would be the graph of $$f^{-1}(x)$$? (All $$x$$ and $$y$$ scales are $$1$$.)

(A) $$\quad\quad$$ (B) $$\quad\quad$$
(C) $$\quad\quad$$ (D) $$\quad\quad$$
(E) Problem 17. $$8e^{\ln 3- \ln 2}=$$

(A) $$8$$ $$\quad\quad$$ (B) $$\frac{e^{\frac32}}8$$ $$\quad\quad$$ (C) $$48$$ $$\quad\quad$$ (D) $$18$$ $$\quad\quad$$ (E) $$12$$

Problem 18. Evaluate $$\frac{d}{dx}\int_{-2}^x\ln t\,dt$$

(A) $$\ln x$$ $$\quad\quad$$ (B) $$\frac1x$$ $$\quad\quad$$ (C) $$\frac1t$$ $$\quad\quad$$ (D) $$\frac1x+\frac12$$ $$\quad\quad$$ (E) $$\ln x-\ln2$$

Problem 19. Select the correct mathematical formulation of the following problem. A farmer wishes to construct $$4$$ adjacent fields alongside a river as shown. Each field is $$x$$ feet wide and $$y$$ feet long. No fence is required along the river, so each field is fenced along $$3$$ sides. The total area enclosed by all $$4$$ fields combined is to be $$800$$ square feet. What is the least amount of fence required? (A) Minimize $$xy$$ if $$4x+5y=800$$
(B) Minimize $$xy$$ if $$4x+5y=200$$
(C) Minimize $$4x+5y$$ if $$xy=800$$
(D) Minimize $$4x+5y$$ if $$xy=200$$
(E) Minimize $$4(x+y)$$ if $$xy=800$$

Problem 20. If $$x^2+y^2=24$$, find $$\frac{d^2y}{dx^2}$$.

(A) $$-\frac xy$$ $$\quad\quad$$ (B) $$-\frac1y$$ $$\quad\quad$$ (C) $$-\frac{24}{y^3}$$ $$\quad\quad$$ (D) $$-2$$ $$\quad\quad$$ (E) $$-1$$

Problem 21. Find the number of units that will minimize the average cost function if the total cost function is $$C(x)=\frac{x^2}4-3x+400$$

(A) $$6$$ $$\quad\quad$$ (B) $$10$$ $$\quad\quad$$ (C) $$20$$ $$\quad\quad$$ (D) $$40$$ $$\quad\quad$$ (E) $$80$$

Problem 22. Find the derivative of $$y=\ln\left(2x+3\right)^7$$

(A) $$14\ln(2x+3)^6$$ $$\quad\quad$$ (B) $$14\ln(2x+3)^7$$ $$\quad\quad$$ (C) $$\frac{14(2x+3)^6}{\ln(2x+3)^7}$$ $$\quad\quad$$ (D) $$\frac{14}{2x+3}$$ $$\quad\quad$$ (E) $$\frac{7}{2x+3}$$

Problem 23. The position of an object at time $$t$$ is given by $$s(t)=2t^3-54t+110$$. Find the acceleration when $$t=2$$

(A) $$30$$ $$\quad\quad$$ (B) $$3$$ $$\quad\quad$$ (C) $$18$$ $$\quad\quad$$ (D) $$24$$ $$\quad\quad$$ (E) $$\frac12$$

Problem 24. Find the average value of $$f(x)=x^2+x$$ on the interval $$[0,1]$$.

(A) $$\frac76$$ $$\quad\quad$$ (B) $$\frac23$$ $$\quad\quad$$ (C) $$1$$ $$\quad\quad$$ (D) $$\frac12$$ $$\quad\quad$$ (E) $$\frac56$$

Problem 25. When expressed as a single logarithm, $$3\log(x-1)-\frac12\log(y+1)-2\log x+\frac14\log z$$ is equal to:

(A) $$\log\frac{(x-1)^3\sqrt{y+1}}{x^2\cdot\sqrtz}$$ $$\quad\quad$$ (B) $$\log\frac{(x-1)^3\sqrt{y+1}^4\sqrt z}{x^2}$$ $$\quad\quad$$ (C) $$\log\frac{(x-1)^3\cdot\sqrtz}{x^2\cdot\sqrt{y+1}}$$ $$\quad\quad$$
(D) $$\log(x-1)^3\sqrtzx^2\sqrt{y+1}$$ $$\quad\quad$$ (E) none of these