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MTH 2205: Final Practice 2 (Part 1: No calculators)

Problem 1. Find the absolute minimum value of \(f(x)=\frac{2x}{x^2+1}\) on the interval \(0\leq x\leq 2\).

(A) \(-1\) \(\quad\quad\) (B) \(0\) \(\quad\quad\) (C) \(1\) \(\quad\quad\) (D) \(\frac45\) \(\quad\quad\) (E) \(2\)

Problem 2. The functions \(f(x)\) and \(f^{-1}(x)\) are inverses of one another, where \(f(-1)=5\), \(f'(5)=7\), and \(f'(-1)=4\). Then \(\left(f^{-1}\right)'(5)\) is

(A) \(-4\) \(\quad\quad\) (B) \(\frac15\) \(\quad\quad\) (C) \(\frac17\) \(\quad\quad\) (D) \(\frac14\) \(\quad\quad\) (E) \(-5\)

Problem 3. The cost of producing \(x\) items is \(C(x)=2x^3-20x^2+100x\). Find the value of \(x\) that minimizes the average cost.

(A) \(5\) \(\quad\quad\) (B) \(6\) \(\quad\quad\) (C) \(3\) \(\quad\quad\) (D) \(8\) \(\quad\quad\) (E) \(2\)

Problem 4. Solve for \(x\): \(\left(\frac13\right)^{1-x}=9\).

(A) \(-2\) \(\quad\quad\) (B) \(0\) \(\quad\quad\) (C) \(-1\) \(\quad\quad\) (D) \(2\) \(\quad\quad\) (E) \(3\)

Problem 5. Solve for \(x\): \(\log_2(x)+\log_2(x-7)=3\).

(A) \(x=1\) and \(x=8\) \(\quad\quad\) (B) \(x=-1\) only \(\quad\quad\) (C) \(x=8\) only \(\quad\quad\)
(D) \(x=3\) and \(x=4\) \(\quad\quad\) (E) \(x=-1\) and \(x=8\)

Problem 6. If \(f(x)=xe^{x^2+1}\), then \(f'(2)\) is

(A) \(9e^5\) \(\quad\quad\) (B) \(6e^5\) \(\quad\quad\) (C) \(2e^5\) \(\quad\quad\) (D) \(8e^5\) \(\quad\quad\) (E) \(e^4\)

Problem 7. \(\int_0^4\frac1{3x+1}\,dx=\)

(A) \(\frac1{13}\) \(\quad\quad\) (B) \(\frac1{13}\ln4\) \(\quad\quad\) (C) \(\frac1{3}\ln13\) \(\quad\quad\) (D) \(\frac14\) \(\quad\quad\) (E) \(\frac3{3x+1}\)

Problem 8. Find \(\frac{dy}{dx}\) for \(y=\ln\left(\frac{\sqrt{x-5}}{4x^3+1}\right)\).

(A) \(\frac{\sqrt{x-5}}{4x^3+1}\) \(\quad\quad\) (B) \(\frac1{2(x-5)}-\frac{12x^2}{4x^3+1}\) \(\quad\quad\) (C) \(\frac{4x^3+1}{\sqrt{x-5}}\) \(\quad\quad\)
(D) \(e^{\sqrt{x-5}}-e^{4x^3+1}\) \(\quad\quad\) (E) \(\frac1{2\sqrt{x-5}}-\frac{12x^2}{4x^3+1}\)

Problem 9. If the second derivative of \(f(x)\) is \(f''(x)=(3-x)(x^2-4)\), on what interval(s) is \(f(x)\) concave up?

(A) \((-\infty,-2)\) and (2,3) \(\quad\quad\) (B) \((-2,2)\) \(\quad\quad\) (C) \((-\infty,\infty)\) \(\quad\quad\)
(D) \((2,\infty)\) \(\quad\quad\) (E) \((-2,2)\) and \((3,\infty)\)

Problem 10. The graph of \(f'(x)\), the derivative of \(f\), is given below for \(-6\leq x\leq 7\). On what intervals is the function, \(f(x)\), increasing?

(A) \((-3,2)\) \(\quad\quad\) (B) \((-3,7)\) \(\quad\quad\) (C) \((-5,-1)\) and \((5,7)\) \(\quad\quad\)
(D) \((-6,-3)\) and \((2,7)\) \(\quad\quad\) (E) \((-6,-5)\) and \((-1,5)\)

Problem 11. The graph of \(y=f(x)\) is given below. Evaluate \(\int_0^4f(x)\,dx\).

(A) \(0.5\) \(\quad\quad\) (B) \(1.5\) \(\quad\quad\) (C) \(-0.5\) \(\quad\quad\) (D) \(3.5\) \(\quad\quad\) (E) \(-2.5\)

Problem 12. An object moving on a line has velocity given by \(v(t)=3t^2-4t+6\), \(t\geq 0\). At time \(t=1\) the object's position is \(s(1)=2\). Find \(s(t)\), the object's position at any time \(t\).

(A) \(s(t)=t^3-2t^2+6t+2\) \(\quad\quad\) (B) \(s(t)=6t-4\) \(\quad\quad\) (C) \(s(t)=t^3-2t^2+6t-3\) \(\quad\quad\)
(D) \(s(t)=t^4-2t^3+6t^2-3\) \(\quad\quad\) (E) \(s(t)=5\)

Problem 13. Linearize \(f(x)=x^4\) at \(x=2\), and then use the linearization to approximate \(f(2.01)\).

(A) \(16.01\) \(\quad\quad\) (B) \(16.24\) \(\quad\quad\) (C) \(18.24\) \(\quad\quad\) (D) \(16.32\) \(\quad\quad\) (E) \(16.04\)

Problem 14. If \(f(x)=\frac{4x-1}{2x+3}\), then the inverse function \(f^{-1}(x)\) is

(A) \(\frac{2x+3}{4x-1}\) \(\quad\quad\) (B) \(\frac{1-4x}{3-2x}\) \(\quad\quad\) (C) \(\frac{4-2x}{3x+1}\) \(\quad\quad\)
(D) \(\frac{3x+1}{4-2x}\) \(\quad\quad\) (E) \(3x+1+\frac1{4-2x}\)

Problem 15. The function \(f(x)=4x^3+9x^2+6x-5\) has a point of inflection at

(A) \(x=1\) \(\quad\quad\) (B) \(x=-\frac12\) \(\quad\quad\) (C) \(x=\frac14\) \(\quad\quad\)
(D) \(x=-\frac34\) \(\quad\quad\) (E) \(x=-\frac12\) and \(x=-1\)

Problem 16. Evaluate \(\displaystyle \int_1^4\left(3\sqrt x+\frac{4}{x^2}\right)\,dx\)

(A) \(6\) \(\quad\quad\) (B) \(7\) \(\quad\quad\) (C) \(18\) \(\quad\quad\) (D) \(10\) \(\quad\quad\) (E) \(17\)

Problem 17. Evaluate \(\displaystyle \sum_{i=1}^4i(4-i)\)

(A) \(0\) \(\quad\quad\) (B) \(-10\) \(\quad\quad\) (C) \(10\) \(\quad\quad\) (D) \(9\) \(\quad\quad\) (E) \(-9\)

Problem 18. \(\displaystyle \int \left(x^3+2x\right)^5\left(12x^2+8\right)\,dx\)

(A) \(\frac16\left(x^3+2x\right)^6+C\) \(\quad\quad\) (B) \(\frac16\left(x^3+2x\right)^6\cdot\frac12\left(12x^2+8\right)^2+C\) \(\quad\quad\) (C) \(\frac12\left(12x^2+8\right)^2+C\) \(\quad\quad\)
(D) \(\frac23\left(x^3+2x\right)^6+C\) \(\quad\quad\) (E) \(\left(x^3+2x\right)^6+C\)

Problem 19. Solve the differential equation \(\frac{dy}{dx}=\frac{x^2}{y^2}\), \(y\neq 0\), with the initial condition \(y(0)=2\).

(A) \(y=\sqrt{x^2+8}\) \(\quad\quad\) (B) \(y=x+8\) \(\quad\quad\) (C) \(y=\sqrt{x^3+1}\) \(\quad\quad\)
(D) \(y=\sqrt[3]{x^3+1}\) \(\quad\quad\) (E) \(y=\sqrt[3]{x^3+8}\)

Problem 20. The demand equation is \(x+3p^2=1000\), where \(p\) is the price. Find the elasticity of demand if \(p=10\).

(A) \(-\frac49\) \(\quad\quad\) (B) \(-\frac16\) \(\quad\quad\) (C) \(-2\) \(\quad\quad\) (D) \(-\frac67\) \(\quad\quad\) (E) \(-\frac58\)

Problem 21. The difference of one number \(x\) and twice a second number \(y\) is \(16\). What is the minimum possible product of \(x\) and \(y\)?

(A) \(-32\) \(\quad\quad\) (B) \(-24\) \(\quad\quad\) (C) \(80\) \(\quad\quad\) (D) \(12\) \(\quad\quad\) (E) \(-48\)

Problem 22. The function \(f(x)=2x^3+3x^2-36x\) has a relative maximum at

(A) \(x=-3\) \(\quad\quad\) (B) \(x=-2\) \(\quad\quad\) (C) \(x=0\) \(\quad\quad\) (D) \(x=2\) \(\quad\quad\) (E) \(x=3\)

Problem 23. The approximate area bounded by \(y=5-x^2\), the \(x\)-axis, \(x=-1\) and \(x=2\), using three rectangles of equal width and right hand endpoints is:

(A) \(9\) \(\quad\quad\) (B) \(10\) \(\quad\quad\) (C) \(11\) \(\quad\quad\) (D) \(12\) \(\quad\quad\) (E) \(13\)

Problem 24. If \(f(2)=7\), \(f'(2)=0\), and \(f''(2)=-5\), then the point \((2,7)\) is

(A) a relative maximum \(\quad\quad\) (B) a relative minimum \(\quad\quad\) (C) a point of inflection \(\quad\quad\)
(D) a point of discontinuity \(\quad\quad\) (E) none of the above

Problem 25. Find the average value of \(f(x)=4x-x^2\) on the interval \(0\leq x\leq 2\).

(A) \(\frac{16}3\) \(\quad\quad\) (B) \(2\) \(\quad\quad\) (C) \(4\) \(\quad\quad\) (D) \(8\) \(\quad\quad\) (E) \(\frac83\)