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

Problem 1. If \(y=x^{\frac32}\), then \(dy\), the differential of \(y\), as \(x\) changes from \(64\) to \(64.1\) is given by

(A) \(12\) \(\quad\quad\) (B) \(1.2\) \(\quad\quad\) (C) \(51.2\) \(\quad\quad\) (D) \(1.5\) \(\quad\quad\) (E) \(0.1\)

Problem 2. Find the critical numbers of \(f(x)=\sqrt{4-x^2}\).

(A) \(x=-4,0,4\) \(\quad\quad\) (B) \(x=-4,4\) \(\quad\quad\) (C) \(x=-2,2\) \(\quad\quad\) (D) \(x=-2,0,2\) \(\quad\quad\) (E) There are no critical numbers

Problem 3. If, for all real numbers \(x\), \(f'(x)>0\) and \(f''(x)<0\), which of the following curves could be a part of the graph of \(f(x)\)?

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

Problem 4. If \(f(x)=x\ln\left(x^2\right)\), then \(\frac{dy}{dx}=\)

(A) \(\ln\left(x^2\right)+2\) \(\quad\quad\) (B) \(\frac1{x^2}\) \(\quad\quad\) (C) \(\frac2x\) \(\quad\quad\) (D) \(\ln\left(x^2\right)+\frac1x\) \(\quad\quad\) (E) \(4\ln(x)\)

Problem 5. The approximate area between the graph of \(y=x^2+2\) and the \(x\)-axis between \(x=1\) and \(x=4\), using \(3\) rectangles of equal width and the RIGHT-HAND endpoints, is:

(A) \(35\) \(\quad\quad\) (B) \(38\) \(\quad\quad\) (C) \(20\) \(\quad\quad\) (D) \(14\) \(\quad\quad\) (E) \(29\)

Problem 6. \(\int\frac{6x^2+8x+2}{x^3+2x^2+x}\,dx=\)

(A) \(\frac{12x+8}{3x^2+4x+1}+C\) \(\quad\quad\) (B) \(\ln\left|x^3+2x^2+x\right|+C\) \(\quad\quad\) (C) \(\frac{2x^3+4x^2+2x}{\frac{x^4}4+\frac{2x^3}3+\frac{x^2}2}+C\) \(\quad\quad\)
(D) \(\frac12\ln\left|x^3+2x^2+x\right|+C\) \(\quad\quad\) (E) \(2\ln\left|x^3+2x^2+x\right|+C\)

Problem 7. The demand curve for a product is given by \(x=1000-4p^2\), where \(p\) is the price. Find the elasticity of demand if \(p=10\).

(A) \(\frac{-3}{4}\) \(\quad\quad\) (B) \(\frac{-4}{3}\) \(\quad\quad\) (C) \(\frac{-1}{4800}\) \(\quad\quad\) (D) \(-2\) \(\quad\quad\) (E) \(\frac{-1}2\)

Problem 8. The average value of \(f(x)=\frac1{\sqrt x}\) on the closed interval \([1,4]\) is

(A) \(\frac{-1}{6}\) \(\quad\quad\) (B) \(\frac{1}{6}\) \(\quad\quad\) (C) \(6\) \(\quad\quad\) (D) \(\frac83\) \(\quad\quad\) (E) \(\frac23\)

Problem 9. For what value of \(k\) will \(y=3x^2+kx-5\) have a minimum at \(x=-2\)?

(A) \(k=10\) \(\quad\quad\) (B) \(k=0\) \(\quad\quad\) (C) \(k=12\) \(\quad\quad\) (D) \(k=\frac72\) \(\quad\quad\) (E) Does not have a minimum value

Problem 10. An object moving on a line has velocity given by the equation \(v(t)=3t^2+t\), for \(t\geq 0\). At time \(t=2\), the object's position is \(s(2)=3\). Find the function describing the position, \(s(t)\), at any time \(t\).

(A) \(s(t)=t^3+\frac12t^2-7\) \(\quad\quad\) (B) \(s(t)=6t-9\) \(\quad\quad\) (C) \(s(t)=t^3+\frac12t^2\) \(\quad\quad\)
(D) \(s(t)=3t^2+t-11\) \(\quad\quad\) (E) \(s(t)=6t+1\)

Problem 11. The function \(f(x)=x^3-6x^2+9x-4\) has a relative maximum at

(A) \(x=0\) \(\quad\quad\) (B) \(x=1\) \(\quad\quad\) (C) \(x=2\) \(\quad\quad\) (D) \(x=3\) \(\quad\quad\) (E) \(x=4\)

Problem 12. If \(f(x)=e^{\frac2x}\), then \(f'(x)=\)

(A) \(2e^{\frac2x}\ln(x)\) \(\quad\quad\) (B) \(e^{\frac2x}\) \(\quad\quad\) (C) \(e^{-\frac{2}{x^2}}\) \(\quad\quad\) (D) \(-\frac2{x^2}e^{\frac2x}\) \(\quad\quad\) (E) \(-2x^2e^{\frac{2}{x}}\)

Problem 13. Let \(f(x)=x^5+x\) and let \(g(x)\) be the inverse of \(f(x)\). Then \(g'(2)=\)

(A) \(-\frac16\) \(\quad\quad\) (B) \(\frac16\) \(\quad\quad\) (C) \(\frac1{81}\) \(\quad\quad\) (D) \(6\) \(\quad\quad\) (E) \(81\)

Problem 14. A window is to be made with a frame as shown. If the total length of the frame is to be \(120\) feet, which of the following should be solved to maximize the area of the window?

(A) Maximize \(3x+2y\), subject to the constraint \(xy=120\)
(B) Maximize \(xy\), subject to the constraint \(3x+2y=120\)
(C) Maximize \(x+y\), subject to the constraint \(3x+2y=120\)
(D) Minimize \(xy\), subject to the constraint \(3x+2y=120\)
(E) Minimize \(3x+2y\), subject to the constraint \(xy=120\)

Problem 15. Find the absolute maximum value of \(f(x)=x^3-3x^2\) on the interval \(-1\leq x\leq 1\).

(A) \(-4\) \(\quad\quad\) (B) \(4\) \(\quad\quad\) (C) \(0\) \(\quad\quad\) (D) \(-2\) \(\quad\quad\) (E) \(2\)

Problem 16. The solution to the differential equation \(\frac{dy}{dx}=\frac{3x}{y}\), \(y\neq 0\), with the initial condition \(y(0)=1\) is:

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

Problem 17. The graph of \(f'\) (the derivative of the function \(f\)) is shown below for \(-4\leq x\leq 5\). On which intervals is the graph of the function, \(f\), increasing?

(A) \(-1 < x< 3\) only \(\quad\quad\) (B) \(-3< x< 1\) and \(3< x< 5\) \(\quad\quad\) (C) \(-3 < x <1\) and \(4< x <5\) \(\quad\quad\)
(D) \(-4< x< -3\) and \(1< x< 4\) \(\quad\quad\) (E) \(-3< x< 1\) only

Problem 18. Solve the following logarithmic equation for \(x\): \(\log_2(x)+\log_2(x+2)=3\)

(A) \(x=-4,2\) \(\quad\quad\) (B) \(x=2\) only \(\quad\quad\) (C) \(x=-3,1\) \(\quad\quad\)
(D) \(x=1\) only \(\quad\quad\) (E) No solution

Problem 19. The graph of \(y=f(x)\) is shown below. Given that the area of the region \(A\) is \(1\) and the area of the region \(B\) is \(3\) and \(\int_{-1}^3f(x)\,dx=\frac32\), what is \(\int_3^2f(x)\,dx\)?

(A) \(-\frac12\) \(\quad\quad\) (B) \(-\frac52\) \(\quad\quad\) (C) \(-1\) \(\quad\quad\) (D) \(1\) \(\quad\quad\) (E) \(-\frac14\)

Problem 20. Evaluate \(\displaystyle \sum_{i=2}^5i(i+1)\).

(A) \(58\) \(\quad\quad\) (B) \(68\) \(\quad\quad\) (C) \(70\) \(\quad\quad\) (D) \(30\) \(\quad\quad\) (E) \(24\)

Problem 21. If \(f(x)=\frac{x+2}x\), find the inverse function, \(f^{-1}(x)\).

(A) \(f^{-1}(x)=x+2\ln x\) \(\quad\quad\) (B) \(f^{-1}(x)=\frac2{x+1}\) \(\quad\quad\) (C) \(f^{-1}(x)=\frac2x\) \(\quad\quad\)
(D) \(f^{-1}(x)=\frac{x}{x+2}\) \(\quad\quad\) (E) \(f^{-1}(x)=\frac2{x-1}\)

Problem 22. If \(f(x)=e^x\) and \(g(x)=\ln(x+1)\), then \(f(g(x))=\)

(A) \(\ln\left(e^x+1\right)\) \(\quad\quad\) (B) \(x(x+1)\) \(\quad\quad\) (C) \(e^x\ln(x+1)\) \(\quad\quad\)
(D) \(\frac{1}{\ln(x+1)}\) \(\quad\quad\) (E) \(x+1\)

Problem 23. The graph of \(y=x^4-6x^3+12x^2+4x+4\) is concave downward whenever

(A) \(-\infty < x< +\infty\) \(\quad\quad\) (B) \(1 < x< 2\) \(\quad\quad\) (C) \(-\infty< x < 1\) or \(2< x< +\infty\) \(\quad\quad\)
(D) \(x< 1\) only \(\quad\quad\) (E) \(x > 2\) only

Problem 24. \(\displaystyle \int_0^1e^{2x}\,dx=\)

(A) \(\frac{e^2-1}2\) \(\quad\quad\) (B) \(\frac{e^2}2\) \(\quad\quad\) (C) \(e^2\) \(\quad\quad\) (D) \(e^2-1\) \(\quad\quad\) (E) \(\frac{e^3-1}2\)

Problem 25. Let \(f(x)\) bet the function defined by \(f(x)=\sqrt{4+x}\). Linearize \(f(x)\) near \(x=0\), and then use this to approximate \(f(0.12)\).

(A) \(2.3\) \(\quad\quad\) (B) \(2.03\) \(\quad\quad\) (C) \(2.04\) \(\quad\quad\) (D) \(4.03\) \(\quad\quad\) (E) \(0.37\)