# Calculus Questions

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Calculus
Application of derivatives
A rectangular storage container with an open top is to have a volume of 60 cubic meters The length of its base is twice the width Material for the base costs 60 dollars per square meter Material for the sides costs 6 dollars per square meter Find the cost of materials for the cheapest such container Minimum cost
Calculus
Application of derivatives
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle What is the area of the largest possibe Norman window with a perimeter of 44 feet
Calculus
Application of derivatives
2200cm of material is available to make a box with a square base and an open top find the largest possible volume of the box Volume cm
Calculus
Application of derivatives
Find the length L and width W with W L of the rectangle with perimeter 44 that has maximum area and then find the maximum area L W Maximum area
Calculus
Application of derivatives
How would you divide a 16 inch line into two parts of length A and B so that A B 16 and the product AB is maximized Assume that A B A B
Calculus
Application of derivatives
se the graph of f x to discuss the graph of y f x Organize your conclusions in a table and sketch a possible graph of y f x Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f has inflection points at x Use a comma to separate answers as needed OB The function f has no inflection points Choose the correct graph of y f x below OA M OB 1 GEXER G OC A Q OD MD
Calculus
Application of derivatives
Antua 9 h D Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h OG a b d e g h E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g O B a b e g OD e g OF a c c d e g OH f x 0 over a h OB a c c e g h OD b c c e g h OF a b d e g h
Calculus
Application of derivatives
Use the graph of y f x assuming f x 0 if x b ord and f x 0 if x 1 to answer parts A through F ANTA E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h C a b e g CHOKES OE e g OG a c c d e g F Identify intervals on which f x is decreasing Choose the correct answer below OA d e g h OC b c c d e g OE a c c d e g OG b c c e g h OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h OB a b e g OD a c c e g h OF e g OH f x is never decreasing on a h
Calculus
Application of derivatives
Use the graph of y f x assuming f x 0 if x b ord and f x 0 if x f to answer parts A through F ANTA AM a E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g OG a c c d e g F Identify intervals on which f x is decreasing Choose the correct answer below OA d e g h OC b c c d e g OE a c c d e g OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h OB a b e g OD a c c e g h OF e g
Calculus
Application of derivatives
Use the graph of y f x assuming f x 0 if x b or d and f x 0 if x f to answer parts A through F AVINA GULB D Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h OG a b d e g h E Identify intervals sol which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g OG a c c d e g OB a b e g OD e g OF a c c d e g OH f x 0 over a h OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h
Calculus
Limits & Continuity
Calculus
Application of derivatives
Use the graph of y f x assuming f x 0 if x b ord and f x 0 if x f to answer parts A through F AVD A D Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h OG a b d e g h E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g OG a c c d e g SLILA food in de Lalain OB a b e g OD e g OF a c c d e g OH f x 0 over a h OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h
Calculus
Application of derivatives
graph of y f x Organize your conclusions in a table and sketch a possible graph of y f x OB The function f is never concave upward Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f has inflection points at x k Use a comma to separate answers as needed OB The function f has no inflection points Choose the correct graph of y f x below OA M OB SAL A x O C Ax O D
Calculus
Application of derivatives
In Silicon Valley a number of computer firms were found to be contaminating underground water supples with toxic che containers A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination Suppose that the required monetary pool in millions of dollars is given by the following function where x is the percentage expressed as a decimal fraction of the total contaminant removed Complete parts A through E P x 3x 1 x 0 x 1 GRZEN OC P x is concave downward on C Find any horizontal and vertical asymptote Find the vertical asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice OA The vertical asymptote s is are x Simplify your answer Use a comma to separate answers as needed OB There are no vertical asymptotes Find the horizontal asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice OA The horizontal asymptote s is are y Simplify your answer Use a comma to separate answers as needed OB There are no horizontal asymptotes
Calculus
Application of derivatives
Calculus
Application of derivatives
Calculus
Vector Calculus
In Silicon Valley a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination Suppose that the required monetary pool in millions of dollars is given by the following function where x is the percentage expressed as a decimal fraction of the total contaminant removed Complete parts A through E P x 3x 1 x 0 x 1 A Where is P x increasing Decreasing Select the correct choice below and fill in the answer box es to complete your choice Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OA P x is increasing on OB P x is increasing on OC P x is decreasing on and decreasing on B Where is the graph of P concave upward Downward Select the correct choice below and fill in the answer box es to complete your choice Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate pooded
Calculus
Vector Calculus
A car rental agency rents 220 cars per day at a rate of 40 per day For each 1 increase in rate 5 fewer cars are rented At what rate should the cars be rented to produce the maximum income What is the maximum income The rental agency will eam a maximum income of MILD when it charges per day
Calculus
Vector Calculus
A company manufactures a variety of camp stoves at different locations The total cost C x in dollars of producing x camp stoves per week at plant A is shown in the figure Discuss the graph of the marginal cost function C x and interpret the graph of C x in terms of the efficiency of the production process at this plant www C x 100 000 50 000 500 1 000 Choose the correct answer below OA The graph of y C x shows that the marginal cost function is positive and decreasing Since marginal costs are decreasin the production process is becoming more efficient as production increases OB The graph of y C x shows that the marginal cost function is negative and increasing Since marginal costs are increasing the production process is becoming less efficient as production increases OC The graph of y C x shows that the marginal cost function is positive and increasing Since marginal costs are increasing the production process is becoming less efficient as production increases OD The graph of y C x shows that the marginal cost function is negative and decreasing Since marginal costs are decreasin the production process is becoming more efficient as production increases
Calculus
Application of derivatives
Use the graph of y f x assuming f x 0 if x b ord and f x 0 if x f to answer parts A through F AVT 9 h 2 C Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h G a b d e g h OA e g OC a c c d e g OE b c c e g h OG b c c d e g D Identify intervals on which f x 0 Choose the correct answer below OB d e g h OD a c c e g h OF a b e g OH f x 20 over a h OB a b e g D e g OF a c c d e g OH f x 0 over a h
Calculus
Application of derivatives
Calculus
Application of derivatives
Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of f x X 4 your choice OA There is an inflection point at x Type an integer or a decimal Use a comma to separate answers as needed OB There are no inflection points Which graph below shows f x OA OB Q O C Q O D Q
Calculus
Limits & Continuity
Use the graph of y f x assuming f x 0 i x bord and f x 0x f to answer parts A through F Q Q AV VA A Identity intervals on which the graph of t is concave upward Choose the correct answer below OA The graph is never concave upward OB a c c e g h OC d e g h OD b c c e g h OF a b d e g h OE a b e g O G a c c d e g OH e g B Identify intervals on which the graph of f is concave downward Choose the correct answer below OA a c c e g h OC b c c e g h OE d e g h OG b ck c dk e a OB a c c d e g OD a b e g OF e g OH The araph is never concave downward
Calculus
Application of derivatives
Use the given information to sketch the graph off Domain All real x lim f x 3 X 1 x 0 3 f x 6 S 3 0 0 1 x 0 ND 0 6 3 X 00 D 3 6 3 0 3 0 ND 6 CID Choose the correct graph below O A O C OB O D
Calculus
Limits & Continuity
Assume f x is continuous on 0 00 Use the given information to sketch the graph off 1 0 3 6 7 1 2 0 f x X f x 4 22 ND 0 2 ND 0 2 Choose the correct graph below OA 3 0 0 OB 6 20 S CAFK O C Q 5 OD
Calculus
Differential equations
K The figure to the right shows the shape and dimension of a small dam Assuming the water level is at the top of the dam and that the shape of the dam is an isosceles triangle find the total force on the face of the dam OB 0 1000 9 8 27 y 3 dy The total force is 21 45x10 N CIT 18 m 27m Save nended Round to the nearest hundredth Corre
Calculus
Vector Calculus
Drag the choices above into the spaces below A B Q D AY Q U E Ay AY U C F Ay www O 57
Calculus
Vector Calculus
4 Square ABCD is drawn as shown below with the diagonals intersecting at E B A D a State four pairs of equivalent vectors b State four pairs of opposite vectors c State two pairs of vectors whose magnitudes are equal but whose directions are perpendicular to each other
Calculus
Indefinite Integration
K A 30 m long chain hangs vertically from a cylinder attached to a winch Assume there is no friction in the system and that the chain has a density of 6 kg m Use 9 8 m s for the acceleration due to gravity a How much work is required to wind the entire chain onto the cylinder using the winch b How much work is required to wind the chain onto the cylinder if a 25 kg block is attached to the end of the chain a Set up the integral that gives the work required to wind the entire chain onto the cylinder using the winch Use increasing limits of integration 30 7 9 8 y dy Type exact answers
Calculus
Differentiation
Prove the following identity and give the values of x for which it is true cos sin x 1 x Take the derivative of each side of the equation cos sin x 1 x dx dx COC
Calculus
Vector Calculus
Part 1 of 2 A particle travels clockwise on a circular path of diameter R monitored by a sensor on the circle at point P the other endpoint of the diameter on which the sensor lies is Q Let 8 be the angle between the diameter PQ and the line from the sensor to the particle Let c be the length of the chord from the particle s position to Q Complete parts a and b below a Calculate de dc ACIER HW Score 30 43 7 of 23 points O Points 0 of 1 Sensor Particle
Calculus
Definite Integrals
How do u evaluate the volume enclosed by x 2 y 2 4 and z 2sqrt x 2 y 2 and z 0 using spherical
Calculus
Vector Calculus
5 Each of the following vector expressions can be simplified and written as a single vector Write the single vector corresponding to each expression and illustrate your answer with a sketch a PORO RS b PS RQ RS PO
Calculus
Vector Calculus
1 The vectors and y are drawn as shown below Draw a vector equivalent to each of the following X
Calculus
Application of derivatives
Part 1 of 6 Find the derivative of the inverse cosine using the theorem of the derivative of the inverse function The theorem states let f be differentiable and have an inverse a point of I at which f x 0 then f is differentiable at y f x and 7 Ye x Let yo cos xo Using the given formula what is the resulting equation 1 OA cos yo OB cos x O C OB cos yo COS xo sin xo sin Xo sin Xo 1 sin xo HW Score 30 43 7 of 23 points O Points 0 of 1
Calculus
Differentiation
A boat sails directly toward a 150 m skyscraper that stands on the edge of a harbor The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer see figure Complete parts a and b below a Use a trigonometric equation to find an expression for the angle e in terms of x Type an equation 8 tan 1 150 X de What is the rate of change of the angular size when the boat is x 400 m from the building dx rad m Round to five decimal places as needed 150 m
Calculus
Limits & Continuity
sinx1 lim 4 x 0 x 2x 6x 2x 6x How should the given limit be evaluated Select the correct choice below and if necessary fill in the answer box to complete your choice OA Use l Hopital s Rule exactly once to rewrite the limit as lim Xx 0 B Use direct substitution C lim x 0 BURD which is convenient and applicable Use l Hopital s Rule more than once to rewrite the limit in its final form as lim x 0 OD Multiply the expression by a unit fraction to obtain lim x 0 Evaluate the limit e sinx 1 eme Type an exact answer e sinx 12x 12x 12
Calculus
Differentiation
A boat is towed toward a dock by a cable attached to a winch that stands 10 ft above the water level see figure Let 6 be the angle of elevation of the winch and let L be the length of the cable as the boat is towed toward the dock Complete parts a to c below de OC dL b Compute LVL 10 10L 100 de dL When L 50 ft When L 20 ft When L 11 ft when L 50 20 and 11 ft de dL de dL de dL 0 0041 Round to four decimal places as needed 0 0289 Round to four decimal places as needed Round to four decimal places as needed GREY de D dL VL TU 10 LL 100 10 ft
Calculus
Limits & Continuity
Prove that lim 1 X 8 lim X 8 818 In 1 a x 1 x Apply l H pital s Rule by finding the derivative of the numerator and the denominator separately 2 lim ax 1 a x lim 2 X 8 X X lim a x48 Evaluate the limit e for a 0 lim Simplify the expression before finding the limit ax 1 a x X X 1 818 a 2 a 2 A
Calculus
Application of derivatives
What is the hybridization of the central atom in BrCl3 sp sp sp3 sp d
Calculus
Application of derivatives
aluate the following integral 2 812 25 2 812 25 dz dz tan 9z 5
Calculus
Application of derivatives
Given the following information evaluate the other five functions 5 sec 8 3 and 0 0 cos sin A 35 Part 2 of 5 NI 2 ACCO
Calculus
Vector Calculus
Recall the following definition Let 0 be an angle in standard position The reference angle of 0 is the acute angle formed by the terminal side of 0 and the horizontal axis Below express the reference angle for 0 in the same units degrees or radians as 0 The reference angle for 254 is The reference angle for 342 is The reference angle for 220 is The reference angle for 429 is degrees degrees degrees degrees
Calculus
Limits & Continuity
Recall the following definition Let 0 be an angle in standard position The reference angle of 0 is the acute angle formed by the terminal side of and the horizontal axis Below express the reference angle for in the same units degrees or radians as 0 You can enter arithmetic expressions like 210 180 or 3 5 pi The reference angle of 100 is The reference angle of 350 is The reference angle of 4 is 0
Calculus
Application of derivatives
Question The rate of increase of rain in Austin can be modeled with the formula R the amount of total rainfall in inches where t is time in days t using the formula R 3 log 8 O 29 O 100 nearest whole number what is the approximate time of which there is a total of 30 inches of rain O28 10 101 To the
Calculus
Limits & Continuity
For the given values of an angle a find the terminal point P x y on the unit circle a If a then x b If a then x c If a then x d If a then e If a 7 then x f If a 37 then x x and Y and y and y and y and y and y GI
Calculus
Application of derivatives
The pH scale for acidity is defined by pH log 0 H where H is the concentration of hydrogen ions measured in moles per liter M A substance has a hydrogen ion concentration of H 3 8 x 10 M Calculate the pH of the substance
Calculus
Application of derivatives
until 11 A certain radioactive isotope has a half life of 2 27 minutes If we initially have 5632 grams of this substance how many half lives will pass remain Your answer should be a whole number help numbers Part 2 What length of time is needed for this decay process to occur Round your answer to the nearest tenth holn numbers minutes grams
Calculus
Application of derivatives
Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F AY a b ww 561 e 9 h Q F Identify intervals on which f x is decreasing Choose the correct answer below O A a c c e g h OC a c c d e g OE b c c e g h OB a b e g O D e g OF b c c d e g ffy is never docrossing on to h
Calculus
Limits & Continuity
Evaluate the following limit lim sinx cotx X 0 lim sinx cotx X 0 Simplify your answer