MA 16010 Exam 3 Practice Problems

A box with a square base and an open top must have a volume of 4000 cm^3. If the cost of the material used is $1 per cm^2, the smallest possible cost of the box is

$1200

Celeste wants to build a fence to enclose a rectangular area of 600 square feet near a stream. The side of the enclosure next to the stream will be made of a waterproof fencing. The other three sides will be made of wood fencing. The waterproof fencing costs $20 per foot, and the wood fencing costs $10 per foot. What is the minimum total cost for the fencing?

$1200

SheSellsSeaShells is an ocean boutique offering shells and handmade shell crafts on Sanibel Island in Florida. Find the price SheSellsSeaShells should charge to maximize revenue if p(x)=160−2x, where p(x) is the price in dollars at which x shells will be sold per day.

$80

A rectangular box with square base and top is to be constructed using sturdy metal. The volume is to be 16 m^3. The material used for the sides costs $4 per square meter, and the material used for the top and bottom costs $1 per square meter. What is the least amount of money that can be spent to construct the box?

$96

Find the point of inflection of h(x)=xe(^−2x)

(1, 1/e^2)

Find the decreasing and concave down interval(s) of f(x)=x(^e−x).

(1,2)

Find the xx-coordinate of the inflection point of y=e^2x−8x^2.

(1/2)*ln(4)

Consider f(x)=16x3+4x−1 and g(x)=x3−6x2−7x+23. On which interval(s) are both f(x) and g(x) concave up?

(2,∞)

Find the largest open interval(s) where f(x)=4x^5 − 5x^4 is concave upward.

(3/4, ∞)

∫(3x^2−4)/(2√x) dx=

(3/5)√(x^5) − 4√x+C

Find the point on the graph of y=5x+2 that is the closest to the point (0,4).

(5/13, 51/13)

Find the inflection point of y=x^3+3x^2.

(−1,2)

Find the open interval where f(x)=1/2x^4+2x^3 is concave downward.

(−2, 0)

∫(sinx−2cosx)/4 dx=

(−2sinx−cosx)/4+C

Find the open interval where the function f(x)=1/3x^3−3x^2+5x−7 is concave down.

(−∞,3)

Suppose the derivative of f(x) is f′(x)=x^3+27. Determine on which interval f(x) is decreasing.

(−∞,−3)

Let f(x)=−x^3+12x. The y values of the absolute minimum and the absolute maximum of f(x) over the closed interval [−3,5] are respectively:

-65 and 16

A poster is to have an area of 200 square inches with 1 inch margins on the left and right sides, and 2 inch margins on the top and bottom. Varying the dimensions of the poster changes the area of the region inside the margins. What is the maximum area inside the margins?

128 square inches

A (closed) rectangular box with a square base will be built for $48. The material for the top and bottom of the box costs $2 per square foot, and the material for the sides of the box costs $1 per square foot. What is the volume of the largest box that can be made?

16 cubic feet

Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. See figure below. Find x which maximizes the area of this window if the total perimeter is 10 feet.

20/(π + 4) ft

An open-top box with a square base is made using 48 ft^2 of material. Find the maximum possible volume of this box.

32 ft^3

A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 160m of wire at your disposal, what is the largest area you can enclose?

3200m^2

Find the x-coordinate of the absolute maximum of g(x)= x^3−3x^2+12 on the closed interval [−2,4].

4

A particle is moving on a straight line with an initial velocity of 10 ft/sec and an acceleration of a(t)=t√+2, where t is time in seconds and a(t) is in ft/sec^2. What is its velocity after 9 seconds?

46ft/sec

A company would like to make a cylindrical can to hold exactly 250π cm^3. What radius will minimize the amount of material needed to construct the can? Recall the volume of a cylinder is given by V=πr^2h and the surface area is given by A=2πr^2+2πrh

5 cm

Find the x-coordinate of the point on the graph of y=√x+2 that is the closest to the point (3,2).

5/2

A box with a square base and open top is to be made from 300 square inches of material. What is the volume of the largest box that can be made.

500 cubic inches

A circular sector with radius r and angle θθ in radians as shown below has an enclosed area of A=1/2r^2θ. The length of the circular arc is rθ. What is the maximum possible area if the perimeter of such a circular sector is 10?

6.25

∫4(³√(x^2))−2/x dx=

6x^2/3−2ln|x|+C

An evergreen nursery usually sells a certain shrub after 5 years of growth and shaping. The growth rate during those 5 years is approximated by dh/dt=1.4t+8,, where t is the time in years and h is the height in centimeters. The seedlings are 14 centimeters tall when planted. How tall are the shrubs when they are sold?

71.5 cm

For rectangles that have a fixed perimeter of 34, what is the largest possible area?

72.25

A company's marketing department has determined that if their product is sold at the price of p dollars per unit, they can sell q=2800−200p units. Each unit costs $10 to make. What is the maximum profit that the company can make?

800 dollars

Choose the correct statement(s) about the function f(x)=2x^3−9x^2. [I.]f(x) has a relative maximum at x=0. [II.] f(x) has a relative minimum at x=3. [III.] f(x) is concave downward on (−∞,3/2).

All of the statements are true.

Consider the function: f(x)=(x^2+2x+1)/(3x−9) Consider the following statements: I. f does not have a horizontal asymptote. II. f has a vertical asymptote at x=3. III. The x-intercept of f is (−1,0). Which of those statements is/are TRUE?

All of them

Choose the correct statement regarding the asymptotes of f(x). f(x)=(x^2−2x+6)/(x+1)

Horizontal Asymptote: None; Vertical Asymptote: x=−1; Slant Asymptote: y=x−3

Consider the function f(x)=(x^2+3x+2)/(x^2−1). Which of the statements are true? [I.] f has a vertical asymptote at x=1. [II.] f has a horizontal asymptote at y=0.[III.] f has a vertical asymptote at x=−1.[IV.] ff has a horizontal asymptote at y=1.

I and IV

The following graph is of f′(x). Choose the correct statement(s) about f(x). I. On (−2,2), f(x) is increasing. II. On (−∞,−2), f(x) is concave up. III. f(x) has a relative maximum at x=0.

I only

Which of the following is/are true: I. limx→∞ (−12x^8+4x^2)/(6x^5+6)=−∞ II. limx→∞(10x^3+100x+1000)/(−5x^3+x^2+x+1)= −1/2 III. limx→∞(14x^6+3)/(x^7+x^9)=14

I only.

f(x) is a polynomial and f′(2)=0, f′(5)=0 f″(3.5)=0, f″(x)<0 on (−∞,3.5) and f″(x)>0 on (3.5,∞) Which of the following statements are true? I. (2,f(2)) is an inflection point of f(x). II. (3.5,f(3.5)) is an inflection point of f(x). III. f(x) has a relative maximum at x=2. IV. f(x) has a relative minimum at x=5.

Only II, III and IV are true.

Let f(x)f(x) be a polynomial whose derivative is always increasing. Choose the correct statement(s). [I.] f(x) has an inflection point. [II.]f(x) has a relative maximum. [III.] f(x) is always concave up.

Only III is correct.

Choose the correct statement regarding the y values of the absolute maximum and the absolute minimum of f(x)=x^3−3x+10 on the interval of [0,3].

The y values of the absolute maximum and the absolute minimum are 28 and 8 respectively.

Find all the asymptotes of f(x)=(1−2x−x^2)/(x+4).

Vertical: x=−4; Horizontal: NONE; Slant: y=−x+2

Find the absolute extrema of f(x)=2x^3+3x^2−36x on the closed interval [0,4].

absolute minimum: (2,−44); absolute maximum: (4,32)

Evaluate the indefinite integral ∫cscx(cotx−cscx) dx.

cotx−cscx+C

limx→∞ f(x)=∞ is true for which of the following functions?

f(x)=(x−x^2)/(−x+5)

Find the particular solution that satisfies the following differential equation and the initial conditions. f″(x)=3cos(x), f′(0)=4, f(0)=7

f(x)=−3cos(x)+4x+10

Which of the following limits equals −∞?

limx→(−∞x^4+8x)/(x^3+1)

Which of the following limits equals to −∞?

limx→∞((2/x) − (x/6))

∫secx(tanx−secx) dx.

secx−tanx+C

Find the x values at which the inflection points of f(x)=1/4x^4 + 2/3x^3 − 15/2x^2+7 occur.

x=−3 and x=5/3

Which of the following describes all the asymptotes of the function f(x)=(−2x^2−5x+7)/(x+3)?

x=−3, y=−2x+1

Find the slant asymptote of h(x)=(3x^3+11x^2+16x+9)/(x^2+2x+1)

y=3x+5

Solve the initial value problem y″=2+4e^x with y′(0)=1 and y(0)=4.

y=x^2 + 4e^x − 3x

Solve the following initial value problem y′= 1/(x^2) + x, y(2)=1

y=−1/x + x^2/2 − 1/2

Solve the initial value problem y′=2sinx+4 with y(0)=1.

y=−2cosx+4x+3