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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) \(\log\frac{(x-1)^3\sqrt{y+1}}{x^2\cdot\sqrt[4]z}\) \(\quad\quad\) (B) \(\log\frac{(x-1)^3\sqrt{y+1}^4\sqrt z}{x^2}\) \(\quad\quad\) (C) \(\log\frac{(x-1)^3\cdot\sqrt[4]z}{x^2\cdot\sqrt{y+1}}\) \(\quad\quad\)
(D) \(\log(x-1)^3\sqrt[4]zx^2\sqrt{y+1}\) \(\quad\quad\) (E) none of these