textbrokercom X 4." Final Exam X 6 1 Canvas LMS X e MATH140-12141(ONL X 6 BusCalc.pdf X * Course Hero X 7 7 C 0 A NotSecure opentextbookstoreroom/buscaIc/BusCalcvpdf it '1'! * El 0 Suggested Sites ' Gmail > YouTube Disney+ canvas 6 find cards saved on. BusCalc.pdf 120 /272 100% 2.6 Exercises In problems 1 and 2, each quotation is a statement about a quantity of something changing over time. Let f(t) represent the quantity at time t. For each quotation, tell what f represents and whether the rst and second derivatives of f are positive or negative 1. (a) "Unemployment rose again, but the rate of increase is smaller than last month" (b) "Our prots declined again, but at a slower rate than last month" (c) "The population is still rising and at a faster rate than last yeart" (a) "The child's temperature is still rising, but slower than it was a few hours ago." 0)) "The number of whales is decreasing, but at a slower rate than last yeari" (c) "The number of people with the u is rising and at a faster rate than last month." On which intervals is the function in the graph (a) concave up? (b) concave down? On which intervals is the function in graph (a) concave up? (b) concave down? textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Course Hero X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 121 / 272 100% 117 Chapter 2 The Derivative Business Calculus 121 6. Sketch the graphs of functions which are defined and concave up everywhere and which have (a) no roots. (b) exactly 1 root. (c) exactly 2 roots. (d) exactly 3 roots. In problems 7 - 10, a function and values of x so that f'(x) = 0 are given. Use the Second Derivative Test to determine whether each point (x, f(x)) is a local maximum, a local minimum or 118 neither 7. f(x) = 2x3 - 15x2 + 6, x=0, 5. 8. g(x) = x3 - 3x2 - 9x + 7, x=-1, 3. 9 . h (x ) = x4 - 8x2 - 2 , x=-2, 0, 2 . 10. f(x) = x.In(x), x=1/e. 119 11. Which of the labeled points in the graph are inflection points? 12. Which of the labeled points in the graph are inflection points? 120 13. How many inflection points can a (a) quadratic polynomial have? (b) cubic polynomial have? (c) polynomial of degree n have? "0" fotextbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Course Hero X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas ) find cards saved o... Other Bookmarks BusCalc.pdf 122 / 272 100% 118 Chapter 2 The Derivative Business Calculus 122 119 15. Fill in the table with "+", "-", or "0" for the function shown. g ( x) g'( x ) g " ( x ) W N - O / AU In problems 16 -22 , find the derivative and second derivative of each function. 120 16. f(x) = 7x2 + 5x - 3 17. f(x) = (2x - 8)5 18. f(x) = (6x - x2) 10 19. f(x) = x . (3x + 7)5 20. f(x ) = (2x3 + 3)6 21. f(x) = Vx2 + 6x- 1 22 . f (x ) = In (x2 + 4 ) 121 122textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Course Hero X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 133 / 272 100% + 2.7 Exercises 1. Find all of the critical points of the function shown and identify them as local max, local min, or 130 neither. Find the global max and min on the interval. 2. Find all of the critical points of the function shown and identify them as local max, local min, or neither. Find the global max and min on the interval. 131 In problems 3 -8, find all of the critical points and local maximums and minimums of each function. 3. f (x) = x2 +8x+7 4. f(x) =2x2 - 12x + 7 AV 5. f (x) = x3 - 6x2 +5 6. f(x) = (x - 1)2 (x- 3) 7. f(x) = In( x2 - 6x + 11 ) 8. f(x) = 2x3 - 96x + 42 In problems 9-16, find all critical points and global extremes of each function on the given intervals. 132 9. f(x) = x2 - 6x + 5 on the entire real number line. 10. f(x) = 2 - x' on the entire real number line. 11. f(x) = x' - 3x + 5 on the entire real number line. 12. f(x)= x -e* on the entire real number line. 13. f(x) = x2 - 6x +5 on [-2, 5] . 14. f(x) = 2 -x on [-2, 1]. 15. f(x) = x3 - 3x +5 on [-2, 1] . 16. f(x) = x -e* on [ 1, 2] . 133textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Course Hero X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas ) find cards saved o... Other Bookmarks E BusCalc.pdf 141 / 272 - 100% + height (feet) 200- 200 200- 100- 100- 100- 10 6. time (minutes) 6. time (minutes) 7 time (minutes) Functions f Derivatives f' 139 8. In the graphs to the right, match the graphs of the functions with those of their derivatives 9. In the graphs below, match the graphs showing the heights of rockets with those showing their velocities. Height 140 IT . P . D.S . D. KA 141 In problems 10 - 14 , use information from the derivatives of each function to help you graph the function. Find all local maximums and minimums of each function. 10. f(x) = x3 - 3x2 -9x -5 11. g(x) = 2x3 - 15x2+6 12. h(x) =x4 -8x2+3 13. r(t) = 2 x - + 3 2 + 1 14. f( x ) = 142textbrokercom X 4." Final Exam X 6 1 Canvas LMS X e MATH140-12141(ONL X 6 BusCalc.pdf X * Course Hero 7 r C' 0 A Not Secure opentextbookstoreloom/buscaIc/BusCalchdf ) '1'! Suggested Sites ' Gmail > YouTube Disney+ canvas 6 find cards saved on. BusCalc.pdf 148 / 272 l. (a) You have 200 feet of fencing available to construct a rectangular pen with a fence divider down the middle (see below) What dimensions of the pen enclose the largest total area? (b) If you need 2 dividers, What dimensions of the pen enclose the largest area? (c) What are the dimensions in parts (a) and (b) if one edge of the pen borders on a river and does not require any fencing? 2. You have 120 feet of fencing to construct a pen with 4 equal sized stalls. If the pen is rectangular and shaped like the one below, what are the dimensions of the pen of largest area and what is that area? D333 , Suppose you decide to fence the rectangular garden in the comer of your yard. Then two sides of the garden are bounded by the yard fence which is already there, so you only need to use the 80 feet of fencing to enclose the other two sides. What are the dimensions of the new garden of largest area? What are the dimensions of the rectangular garden of largest area in the corner of the yard if you have F feet of new fencing available? i (a) You have a 10 inch by 15 inch piece of tin which you plan to form into a box (without a top) by cutting a square from each corner and folding up the sides. How much should you cut from each comer so the resulting box has the greatest volume? (b) If the piece of tin is A inches by B inches, how much should you cut from each corner so the resulting box has the greatest volume? . You have a 10 inch by 10 inch piece of cardboard which you plan to cut and fold as shown to form a box with a top. Find the dimensions of the box which has the largest volume. textbrokercom X 4." Final Exam X 6 1 Canvas LMS X e MATH140-12141(ONL X 6 BusCalc.pdf X * Course Hero X 7 7 C 0 A NotSecure opentextbookstoreroom/buseaIc/BusCalcvpdf it '1'! * El 0 Suggested Sites ' Gmail > YouTube Disney+ canvas 6 find cards saved on. BusCalc.pdf 149 /272 100% + El 03 of cardboard which you plan to cut and fold as shown to form a box with a top. Find the dimensions of the box which has the largest volume. Chapter 2 The Derivative Business Calculus 6. (a) You have been asked to bid on the construction of a squarebottomed box with no top which will hold 100 cubic inches of water. If the bottom and sides are made 'om the same material, what are the dimensions of the box which uses the least material? (Assume that no material is wasted) (1)) Suppose the box in part (a) uses different materials for the bottom and the sides If the bottom material costs 5 per square inch and the side material costs 3 per square inch, what are the dimensions of the least expensive box which will hold 100 cubic inches of water? (a) Determine the dimensions of the least expensive cylindrical can which will hold 100 cubic inches if the materials cost 2, 5 and 3 respectively for the top, bottom and sides. (b) How do the dimensions of the least expensive can change if the bottom material costs more than 53 per square inch? You have 100 feet of fencing to build a pen in the shape of a circular sector, the "pie slice" shown The area of such a sector is rs/Z. What value of r maximizes the enclosed area? textbrokercom X 4." Final Exam X 6 1 Canvas LMS X e MATH140-12141(ONL X 6 BusCaIc.pdf X * Course Hero X 7 7 C 0 A NotSecure opentexrbookstoreloom/buseaIc/BusCalcpdf it '1'! * El 0 Suggested Sites ' Gmail > YouTube Disney+ canvas 6 find cards saved on. BusCalc.pdf 154 / 272 Chapter 2 The Derivative Business Calculus 2.10 Exercises 1. If g(20) : 35 and g'(2o) : 2, estimate the value of g(22). 2. If g(1)= 717 and g'(l)= 5 , estimate the value of g(1.2)i 3. Use the Tangent Line Approximation to estimate the cube root of 9. 4. Use the Tangent Line Approximation to estimate the fih root of 30' 5. A rectangle has one side on the xaxis, one side on the yaxis, and a corner on the graph of y = x2 + 1 . (a) Use Linear Approximation of the area formula to estimate the increase in the area of the rectangle if the base grows from 2 to 23 inches (b) Calculate exactly the increase in the area of the rectangle as the base grows from 2 to 23 inches , You can measure the diameter of a circle to within 0.3 cm. (a) How large is the "error" in the calculated area of a circle with a measured diameter of 7.4 cm? (b) How large is the "error" in the calculated area of a circle with a measured diameter of 13.6 cm? (c) How large is the percentage error in the calculated area of a circle with a measured diameter of d? 7. The demand function for Alicia's oven mitts is given by q : 8 p + 80 (q is the number of oven mitts, p is the price in dollars), Find the elasticity of demand when p = $750 Will revenue increase if Alicia raises her price from $7.50? 8. The demand function for Shaki's danglies is given by q : 35 p + 205 (q is the number of danglies, z; is the price in dollars per danglvt. Find the elasticitv of demand when n = $5 Should Shaki textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Course Hero X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 159 / 272 69% 156 In problems 1 -10 find dy/dx by differentiating implicitly then find the value of dy/dx at the given point. 1. x2 + y = 100, point (6, 8) 2. x2 + 5y? =45, point (5, 2) 3. x2 - 3xy + 7y=5 , point (2,1) 4. Vx + Vy =5, point (4,9) 5. C + Y 5. a + 16 = 1 , point (3,0) 7. In(y) + 3x - 7=0 , point (2,e) B. x2 - y = 16 , point (5,3) 9. x2 - y = 16 , point (5, -3) 10. y + 7x3 - 3x =8 , point (1,2) 157 11. Find the slopes of the lines tangent to the graph in x = 4y-y shown at the points (3,1), (3,3), and (4,2) . 12. Find the slopes of the lines tangent to the graph in shown where the graph crosses the y-axis. 13. Find the slopes of the lines tangent to the graph in graph shown at the points ((5,0), (5,6), and (-4,3). 14. Find the slopes of the lines tangent to the graph in x =y' -6y+5 158 the graph shown where the graph crosses the y-axis. In problems 15-16, find dy/dx using implicit differentiation and then find the slope of the line angent to the graph of the equation at the given point. 15. y' - 5y = 5x2 +7 , point (1,3) 16. y - 5xy + x2 + 21 = 0, point (2,5) 159 Chapter 2 The Derivative Business Calculus 160 17. An expandable sphere is being filled with liquid at a constant rate from a tap (imagine a water balloon connected to a faucet). When the radius of the sphere is 3 inches, the radius is 160 increasing at 2 inches per minute. How fast is the liquid coming out of the tap? ( V =3 3 )
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