%Covers Chapter 4 \magnification=\magstep1 \centerline{\bf Sample Quiz Number 4, Math 1251, Scot Adams, Fall 1998} \vskip.1in\noindent No calculators. 50 minutes. Every problem is worth one point. \vskip.1in\noindent Answers: CGFF EETF GEEF FAETTFFT EEFFTTT FFI %***************** \vskip.2in\noindent 1. Let $f$ be a function whose first derivative is given by $\displaystyle{f'(x)={x+4\over(x-1)^3}}$. Then the {\it complete} set of intervals on which $f$ is decreasing is \vskip.1in\noindent\line{ (A) $(-\infty,-4]$\hfil (B) $(-\infty,-4]$ and $(1,\infty)$\hfil (C) $[-4,1)$}\vskip.1in\noindent\line{ (D) $(-\infty,-3]$\hfil (E) $(-\infty,1]$\hfil (F) $(-\infty,4]$}\vskip.1in\noindent\line{ (G) $(-\infty,-4]$ and $[-1,\infty)$\hfil (H) $[-1,4]$\hfil (I) $(-\infty,3]$}\vskip.1in\noindent\line{\hfil (J) $(-\infty,-1)$\hfil (K) none of these\hfil} %answer C %section 4.3 Monotonic functions and the first derivative test %***************** \vskip.2in\noindent 2. Let $f(x)=x^3+3x^2-9x+8$. Then \vskip.1in\noindent (A) $f$ is increasing on $[0,\infty)$\qquad and is decreasing on $(-\infty,0]$. \vskip.1in\noindent (B) $f$ is increasing on $(-\infty,0]$ and on $[4,\infty)$\qquad and is decreasing on $[0,4]$. \vskip.1in\noindent (C) $f$ is increasing on $(-\infty,0)$\qquad and is decreasing on $[0,\infty)$. \vskip.1in\noindent (D) $f$ is increasing on $[4,\infty)$\qquad and is decreasing on $(-\infty,4]$. \vskip.1in\noindent (E) $f$ is increasing everywhere. \vskip.1in\noindent (F) $f$ is decreasing on $(-\infty,-3]$ and on $[1,\infty)$\qquad and is increasing on $[-3,1]$. \vskip.1in\noindent (G) $f$ is increasing on $(-\infty,-3]$ and on $[1,\infty)$\qquad and is decreasing on $[-3,1]$. \vskip.1in\noindent (H) $f$ is decreasing everywhere. \vskip.1in\noindent (I) none of these. %answer G %section 4.3 Monotonic functions and the first derivative test %***************** \vskip.2in\noindent 3. Let $f(x)=|x-1|$. Then there exists a number $c\in(0,2)$ such that $$f(2)-f(0)=[f'(c)][2-0].$$ \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.2 The mean value theorem %problem 17, p. 267, Steward 3rd edition %***************** \vskip.2in\noindent 4. Let $f(x)=(x+2)^5$. Then $f$ has a point of inflection at the point $(2,0)$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.4 Concavity and points of inflection %***************** \vskip.2in\noindent CONTINUE TO NEXT PAGE \vfil\eject\noindent 5. Let $f(x)=x^7$. Then \vskip.1in\noindent (A) $f$ is concave upward on $(-\infty,-7]$\qquad and is concave downward on $[-7,\infty)$. \vskip.1in\noindent (B) $f$ is concave downward on $(-\infty,-7]$\qquad and is concave upward on $[-7,\infty)$. \vskip.1in\noindent (C) $f$ is concave upward on $(-\infty,-1]$ and on $[1,\infty)$\qquad and is concave downward on $[-1,1]$. \vskip.1in\noindent (D) $f$ is concave downward on $(-\infty,-1]$ and on $[1,\infty)$\qquad and is concave upward on $[-1,1]$. \vskip.1in\noindent (E) $f$ is concave downward on $(-\infty,0]$\qquad and is concave upward on $[0,\infty)$. \vskip.1in\noindent (F) $f$ is concave upward on $(-\infty,0]$\qquad and is concave downward on $[0,\infty)$. \vskip.1in\noindent (G) $f$ is concave upward everywhere. \vskip.1in\noindent (H) $f$ is concave downward everywhere. %answer E %section 4.4 Concavity and points of inflection %***************** \vskip.2in\noindent 6. Let $f(x)=16x^3+12x^2+6x+5$. Then \vskip.1in\noindent (A) $f$ is concave upward on $(-\infty,-1/4]$\qquad and is concave downward on $[-1/4,\infty)$. \vskip.1in\noindent (B) $f$ is concave upward on $[0,\infty)$\qquad and is concave downward on $(-\infty,0]$. \vskip.1in\noindent (C) $f$ is concave upward on $(-\infty,1/4]$\qquad and is concave downward on $[1/4,\infty)$. \vskip.1in\noindent (D) $f$ is concave downward on $(-\infty,1/4]$\qquad and is concave upward on $[1/4,\infty)$. \vskip.1in\noindent (E) $f$ is concave downward on $(-\infty,-1/4]$\qquad and is concave upward on $[-1/4,\infty)$. \vskip.1in\noindent (F) $f$ is concave upward on $(-\infty,0]$\qquad and is concave downward on $[0,\infty)$. \vskip.1in\noindent (G) $f$ is concave upward on $[2,\infty)$\qquad and is concave downward on $(-\infty,2]$. \vskip.1in\noindent (H) $f$ is concave upward on $(-\infty,2]$\qquad and is concave downward on $[2,\infty)$. \vskip.1in\noindent (I) $f$ is concave upward everywhere. %answer E %section 4.4 Concavity and points of inflection %***************** \vskip.2in\noindent 7. Let $f$ be twice differentiable on $(-1,1)$ and assume that $f''(x)>0$, for all $x\in(-1,1)$. Then $f$ is concave upward on $(-1,1)$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.4 Concavity and points of inflection %``The test for concavity'', p. 274, Steward 3rd edition %***************** \vskip.2in\noindent 8. If a function is increasing on an interval, then it must be concave upward on that interval. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.4 Concavity and points of inflection %***************** \vskip.2in\noindent CONTINUE TO NEXT PAGE \vfil\eject\noindent 9. Let $f(x)=x^4+4x^3+4x^2+8$. Then \vskip.1in\noindent (A) $f$ is increasing on $[-\sqrt{3},0]$ and on $[\sqrt{3},\infty)$\qquad and is decreasing on $(-\infty,-\sqrt{3}]$ and on $[0,\sqrt{3}]$. \vskip.1in\noindent (B) $f$ is decreasing on $(-\infty,0]$\qquad and is increasing on $[0,\infty)$. \vskip.1in\noindent (C) $f$ is increasing on $(-\infty,-\sqrt{3}]$ and on $[0,\sqrt{3}]$\qquad and is decreasing on $[-\sqrt{3},0]$ and on $[\sqrt{3},\infty)$. \vskip.1in\noindent (D) $f$ is increasing everywhere. \vskip.1in\noindent (E) $f$ is decreasing everywhere. \vskip.1in\noindent (F) $f$ is increasing on $(-\infty,-2]$ and on $[-1,0]$\qquad and is decreasing on $[-2,-1]$ and on $[0,\infty)$. \vskip.1in\noindent (G) $f$ is decreasing on $(-\infty,-2]$ and on $[-1,0]$\qquad and is increasing on $[-2,-1]$ and on $[0,\infty)$. \vskip.1in\noindent (H) none of these. %answer G %section 4.3 Monotonic functions and the first derivative test %***************** \vskip.2in\noindent 10. The illumination of an object by a light source is equal to $${\hbox{the strength of the source}\over (\hbox{the distance from the source to the object})^2}.$$ Imagine that two light sources are placed on the real number line. One is of strength one and is placed at $0$. The other is of strength eight and is placed at $15$. The object is placed between them. %Note: the ratio of the strengths should be a cube to get a good answer. %In this case the the cube root of 8 is 2, so the answer is x=15/(2+1)=5. %In general you get x=a/(b+1), where a is the position of the second object, %and b is the cube root of the ratio of the strengths. Let $x$ denote the number at which the object should be placed so as to receive the {\it least} illumination. Then $x$ is equal to \vskip.1in\noindent\line{ (A) $1$\hfil (B) $2$\hfil (C) $3$\hfil (D) $4$\hfil (E) $5$\hfil (F) $6$\hfil (G) $10$\hfil (H) $11$\hfil (I) $12$\hfil (J) $13$\hfil (K) $14$}\vskip.1in\noindent\centerline{ (L) none of these} %answer E %section 4.7 Applied maximum and minimum problems %problem 39, p. 300, Steward 3rd edition %***************** \vskip.2in\noindent 11. A farmer with $1200$ feet of fencing wants to enclose a rectangular area and then divide it into two pens with fencing parallel to one side of the rectangle. What is the largest possible total area (in square feet) of the two pens? \vskip.1in\noindent\line{ (A) $(150)(150)$\hfil (B) $(50)(450)$\hfil (C) $(120)(290)$\hfil (D) $(100)(300)$}\vskip.1in\noindent\centerline{ (E) none of these} %answer E %section 4.7 Applied maximum and minimum problems %similar to problem 7, p. 299, Steward 3rd edition %***************** \vskip.2in\noindent 12. The graph of $y=\arcsin x$ has no points of inflection. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.4 Concavity and points of inflection %***************** \vskip.2in\noindent CONTINUE TO NEXT PAGE \vfil\eject\noindent 13. The graph of $y=2x^2+3x+5$ is increasing on $(-\infty,\infty)$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.3 Monotonic functions and the first derivative test %***************** \vskip.2in\noindent 14. Find the critical numbers of $f(x)=xe^{4x}$. \vskip.1in\noindent\line{ (A) $-1/4$ only\hfil (B) $0$ and $1$\hfil (C) $-1/4$ and $0$\hfil (D) $-1/4$ and $1$}\vskip.1in\noindent\centerline{ (E) none of these} %answer A %section 4.1 %similar to problem 40, p. 261, Steward 3rd edition %***************** \vskip.2in\noindent 15. Find the absolute minimum and absolute maximum values of $f(x)=x^2-2x+2$ on $[0,2]$. \vskip.1in\noindent\line{ (A) $1$ and $5$\hfil (B) $1$ and $10$\hfil (C) $5$ and $10$\hfil (D) $2$ and $5$}\vskip.1in\noindent\line{ (E) $1$ and $2$\hfil (F) $1$ and $3$\hfil (G) $0$ and $4$\hfil (H) $0$ and $5$}\vskip.1in\noindent\centerline{ (I) none of these} %answer E %section 4.1 %similar to problem 41, p. 261, Steward 3rd edition %***************** \vskip.2in\noindent 16. If a function is continuous on the interval $[2,3]$, then it must attain a maximum value on $[2,3]$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.1 Maximum and minimum values %The extreme value theorem on p. 255, Steward 3rd edition %***************** \vskip.2in\noindent 17. Let $f$ be a function whose domain is the interval $[1,5]$. Suppose that $f$ attains an absolute maximum at $3$. Then $3$ must be a critical number for $f$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.1 Maximum and minimum values %``Fermat's theorem'' on p. 257, Steward 3rd edition %***************** \vskip.2in\noindent 18. True or False: Assume that $f$ is a function which is differentiable at every real number. Assume also that $f'(3)=0$. Then $f$ attains a local extremum at $3$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.1 Maximum and minimum values %``Fermat's theorem'' on p. 257, Steward 3rd edition %***************** \vskip.2in\noindent 19. Let $x,y,r,\theta$ be defined as in planetary motion. Suppose we prove that $1/r=2+3\sin\theta$. Find $A,B,C,D,E,F$ such that $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ and such that $A=-4$. Then $C$ is equal to \vskip.1in\noindent\line{ (A) 1\hfil (B) 2\hfil (C) 3\hfil (E) 4\hfil (F) 5\hfil (G) 6\hfil (H) 7\hfil (I) 8\hfil (J) none of these} %answer F %section planetary motion %***************** \vskip.2in\noindent 20. True or False: If two functions are both differentiable at every real number, and if they have the same derivative, then they must differ by a constant. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.2 The Mean Value Theorem %***************** \vskip.2in\noindent CONTINUE TO NEXT PAGE \vfil\eject\noindent 21. Recall that $\dot r=dr/dt$ and that $\dot\theta=d\theta/dt$. Compute $[d/dt][r\csc\theta]$. \vskip.1in\noindent\line{ (A) $\dot r\csc\dot\theta$\hfil (B) $\dot r\csc\theta\cot\theta$\hfil (C) $\dot r\csc\theta+r\dot\theta\csc\theta\cot\theta$} \vskip.1in\noindent\line{ (D) $-\dot r\csc\theta+r\dot\theta\csc\theta\cot\theta$\hfil (E) $\dot r\csc\theta-r\dot\theta\csc\theta\cot\theta$\hfil (F) none of these} %answer E %section planetary motion %***************** \vskip.2in\noindent 22. Find the intervals of increase and decrease for $f(x)=x^3$. \vskip.1in\noindent (A) $f$ is increasing on $(-\infty,0]$\qquad and is decreasing on $[0,\infty)$. \vskip.1in\noindent (B) $f$ is decreasing on $(-\infty,0]$\qquad and is increasing on $[0,\infty)$. \vskip.1in\noindent (C) $f$ is increasing on $(-\infty,-1]$ and on $[1,\infty)$\qquad and is decreasing on $[-1,1]$; \vskip.1in\noindent (D) $f$ is decreasing on $(-\infty,-1]$ and on $[1,\infty)$\qquad and is increasing on $[-1,1]$; \vskip.1in\noindent (E) $f$ is increasing everywhere. \vskip.1in\noindent (F) $f$ is decreasing everywhere. \vskip.1in\noindent (G) none of these. %answer E %section 4.3 Monotonic functions and the first derivative test %***************** \vskip.2in\noindent 23. True or False: A local maximum is always an absolute maximum. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.1 Maximum and minimum values %***************** \vskip.2in\noindent 24. True or False: An absolute maximum is always a local maximum. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.1 Maximum and minimum values %***************** \vskip.2in\noindent 25. True or False: If $f$ is increasing on $(-\infty,\infty)$, then $f$ is one-to-one. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.3 Monotonic functions and the first derivative test %***************** \vskip.2in\noindent 26. True or False: If $f$ is a constant function, then $f$ has a local maximum at every real number. (NOTE: The domain of any constant function is the set of all real numbers.) \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.1 Maximum and minimum values %***************** \vskip.2in\noindent 27. True or False: Let $f$ be continuous on the compact interval $[a,b]$. Then $f$ attains an absolute maximum on $[a,b]$ and an absolute minimum on $[a,b]$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer T %section 4.1 Maximum and minimum values %The Extreme Value Theorem, (3) on p. 255 %***************** \vskip.2in\noindent CONTINUE TO NEXT PAGE \vfil\eject\noindent 28. True or False: Let $f$ be continuous on the open interval $(a,b)$. Then $f$ attains an absolute maximum and an absolute minimum on $(a,b)$. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.1 Maximum and minimum values %The Extreme Value Theorem, (3) on p. 255 %***************** \vskip.2in\noindent 29. True or False: A constant function is decreasing. \vskip.1in\noindent\line{ (T) True\hfil (F) False\hfil} %answer F %section 4.1 Maximum and minimum values %***************** \vskip.2in\noindent 30. Compute the components of the gravitational force arrow for a particle of mass $50$ located at the point $(3,-4)$. \vskip.1in\noindent\line{ (A) $\langle3,-4\rangle$\hfil (B) $\langle3/50,-4/50\rangle$\hfil (C) $\langle-3,4\rangle$}\vskip.1in\noindent\line{ (D) $\langle-3/50,4/50\rangle$\hfil (E) $\langle6,-8\rangle$\hfil (F) $\langle-6,8\rangle$}\vskip.1in\noindent\line{ (G) $\langle3/5,4/5\rangle$\hfil (H) $\langle-3/5,4/5\rangle$\hfil (I) none of these} %answer I %section planetary motion %***************** \vskip.2in\noindent END OF TEST \end