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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) $$0.37$$