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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)** \(0.37\)