MSTC LOOKSFAM PART 2 (CALCULUS)

Find all points on the graph of y = |xex | at which the graph has a horizontal tangent.

(-1, 1/e)

Find the coordinates of each point on the graph of y2 - x2 - 6x + 7 = 0 at which the tangent line is vertical. Write an equation of each vertical tangent.

(-7, 0) and (1, 0) ,Tangent lines x = -1, x = 1

Given the cost function C(x) = 100 + 8x + 0.1x2 , (a) find the marginal cost when x = 50; and (b) find the marginal profit at x = 50, if the price per unit is $20.

(a) $18 (b) $2

The velocity function of a moving particle on a coordinate line is v(t) = 3 cos(2t) for 0 ≤ t ≤ 2π. Using a calculator: (a)Determine when the particle is moving to the right.(b)Determine when the particle stops.(c)Determine the total distance traveled by the particle during 0 ≤ t ≤ 2π

(a) 0 < t < π /4 , 3π /4 < t < 5π /4 , 7π/ 4 < t < 2π, (b)t = π/4 , 3π/4 , 5π/4 , and 7π/4, (c)12

If the marginal cost of producing x units of a commodity is C'(x) = 5 + 0.4x , (a)Find the marginal cost when x = 50 (b)Find the cost producing the first 100 units.

(a) 25, (b) 2500

If the amount of bacteria in a culture at any time increases at a rate proportional to the amount of bacteria present and there are 500 bacteria after one day and 800 bacteria after the third day: (a)Approximately how many bacteria are there initially?(b)Approximately how many bacteria are there after 4 days?

(a) 395 (b) 1011

The velocity function of a moving particle on a coordinate line is v(t) = t2 + 3t − 10 for 0 ≤ t ≤ 6. Find (a) the displacement by the particle during 0 ≤ t ≤ 6, and (b) the total distance traveled by the particle during 0 ≤ t ≤ 6.

(a) 66 (b) 266/3

The velocity function of a moving particle on a coordinate line is v(t) = t3 − 6t2 + 11t − 6. Using a calculator, find (a) the displacement by the particle during 1 ≤ t ≤ 4 and (b) the total distance traveled by the particle during 1 ≤ t ≤ 4.

(a) 9/4 (b) 11/4

The position function of a particle moving on a straight line is s (t) = 2t3 − 10t2 + 5. Find (a) the position, (b) instantaneous velocity, (c) acceleration, and (d) speed of the particle at t = 1.

(a) s = -3 (b) v(1) = -14 (c) a(1) = -8 (d) Speed = 14

From a 400-foot tower, a bowling ball is dropped. The position function of the bowling ball s (t) = −16t2 + 400, t ≥ 0 is in seconds. Find: the instantaneous velocity of the ball at t = 2 seconds. (b)the average velocity for the first 3 seconds. (c) when the ball will hit the ground.

(a) v(2) = -64 ft/sec (b) average, v = -48 ft/sec (c) t = 5 sec

Given the curve y = 1/x : (a) write an equation of the normal line to the curve y = 1/x at the point (2, 1/2), and (b) does this normal line intersect the curve at any other point? If yes, find the point.

(a) y = 4x -15/2 (b) (-0.125, -8)

Find the point on the graph of y = ln x such that the normal line at this point is parallel to the line y = −e x − 1.

(e, 1)

If h is the diameter of a circle and h is increasing at a constant rate of 0.1 cm/sec, find the rate of change of the area of the circle when the diameter is 4 cm.

0.2π cm2/sec

Find the area of the region bounded by y = sin x and y = x/2 between 0 ≤ x ≤ π.

0.421

The velocity of a particle moving on a line v(t) = 3t2 - 18t + 24. Find the average velocity from t = 1 to t = 3.

1 units/sec

Two containers are being used. One container is in the form of an inverted right circular cone with height of 10 meters and a radius at the base of 4 meter. The other container is a right circular cylinder with a radius of 6 meters a height of 8 meters. If water is being drained from the conical container in to the cylindrical container at the rate of 15 m /min, how fast is the water level falling in the conical tank when the water level in the conical tank is 5 meters high?

1.19 m/min

Find the area of the regions bounded by the graphs of f (x ) = x3 and g(x) = x .

1/2

Find the area under the curve from x = 0 to x = 2

1/2

Write an equation for each normal line to the graph of y = 2 sin x for 0 ≤ x ≤ 2π that has a slope of 1/2 .

1/2

Find the instantaneous rate of change at x = 5 of the function

1/3

Find the volume of a pyramid whose base is a square with a side of 6 feet long, and a height of 10 feet.

120 ft3

Find the volume of the solid generated by revolving about the line y =−3, the region bounded by the graph of y = ex , the y-axis, and the lines x = ln 2 and y = −3.

13.7383π

Find the volume of the solid generated by revolving about the y-axis the region in the first quadrant bounded by the graph of y = x2 , the y-axis, and the line y = 6.

18π

Find the area of the region bounded by the curve xy = 1 and the lines y = −5, x = e, and x = e3 .

2 - 5e + 5e3

The base of a solid is the region enclosed by a triangle whose vertices are (0, 0), (4, 0), and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Using a calculator, find the volume of the solid.

2.094

Using your calculator, find the shortest distance between the point (4, 0) and the line y = x .

2.828

A light on the ground 100 feet from a building is shining at a 6-foot-tall man walking away from the light and toward the building at the rate of 4 ft/sec. How fast is his shadow on the building becoming shorter when he is 40 feet from the building?

2/3 fps

Find the average value of y = sin x between x = 0 and x = π.

2/π

Water is leaking from a faucet at the rate of l(t) = 10e-0.5t gallons per hour, where t is measured in hours. How many gallons of water will have leaked from the faucet after a 24 hour period?

20 gallons

The base of a solid is the region enclosed by the ellipse . The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid.

200/3

Find the area bounded by f(x) = x3 + x2 − 6x and the x-axis.

21.083

In a farm, the animal population is increasing at a rate which can be approximately represented by g(t) = 20 + 50 ln(2 + t), where t is measured in years. How much will the animal population increase to the nearest tens between 3rd and 5th year?

220

The radius of a sphere is increasing at a constant rate of 2 inches per minute. In terms of the surface area, what is the rate of change of the volume of the sphere?

28 in3/min

Find the area of the region bounded by x = y2 , y = −1, and y = 3.

28/3

Given the cost function C(x) = 500 + 3x + 0.01x2 and the demand function (the price function) p(x) = 10, find the number of units produced in order to have maximum profit.

350 units

Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region bounded by y = x2 and x = y2 about the y- axis.

3π/10

Find the area of the region bounded by the graph of f(x) = x2 − 1, the lines x = −2 and x = 2, and the x-axis.

4

Using the Washer Method, find the volume of the solid generated by revolving the region bounded by y = x3 and y = x in the first quadrant about the x- axis.

4π/21

Find the shortest distance between point A (19,0) and the parabola y = x2 -2x + 1

4√17 units

Find the volume of the solid generated by revolving about the line y = 8, the region bounded by the graph of y = x2 + 4, and the line y = 8.

512π/15

Let f be a continuous function defined on [0, 12] as shown below. Find the Riemann sum for f(x) over [0, 12] with 3 subdivisions of equal length and the midpoints of the intervals as ci. Hint:

596

The acceleration function of a moving particle on a coordinate line is a(t) = −4 and v0 = 12 for 0 ≤ t ≤ 8. Find the total distance traveled by the particle during 0 ≤ t ≤ 8.

68

The population of the Great Britain was 57.1 million in 2001 and 60.6 million in 2006. Find a logistic model for growth of the population, assuming a carrying capacity of 100 million. Use the model to predict the population in 2020.

69.742 million

The marginal profit of manufacturing and selling a certain drug is P(x) = 100 − 0.005x . How much profit should the company expect if it sells 10,000 units of this drug?

750,000

The height of a right circular cone is always three times the radius. Find the volume of the cone at the instant when the rate of increase of the volume is twelve times the rate of increase of the radius.

8/√π or 4.51

Find the volume of the solid generated by revolving about the x -axis the region bounded by the graph of , the x-axis, and the line x = 5.

8π

Find the surface area of a sphere at the instant when the rate of increase of the volume of the sphere is nine times the rate of increase of the radius.

9 sq.units

The spread of an infectious disease can often be modeled by a logistic equation with the total exposed population as the carrying capacity. In a community of 2000 individuals, the first case of a new virus is diagnosed on March 31, and by April 10, there are 500 individuals infected. Write a differential equation that models the rate at which the virus spread through the community and determine when 98% of the population will have contracted the virus.

April 18

The graph of encloses a region with the x -axis and y-axis in the first quadrant. A rectangle in the enclosed region has a vertex at the origin and the opposite vertex on the graph of . Find the dimensions of the rectangle so that its area is a maximum.

Length = 2, width = 1

If the line y = 6x + a is tangent to the graph of y = 2x3, find the value(s) of a.

a = ±4

The velocity function of a moving particle is for 0 ≤ t ≤ 7. What is the minimum and maximum acceleration of the particle on 0 ≤ t ≤ 7

amin = dv/dt = 0 at t = 4, amax = dv/dt = 16 at t = 0

Given , verify the hypotheses of the Mean Value Theorem for Integrals for f on [1, 10] and find the value of c as indicated in the theorem.

c = 2√3

Find the area of the region bounded by the curve y = ex , the y-axis, and the line y = e2 .

e2 + 1

The region bounded by the x -axis, and the graph of y = sin x between x = 0 and x = π is divided into 2 regions by the line x = k. If the area of the region for 0 ≤ x ≤ k is twice the area of the region k ≤ x ≤ π, find k

k = arccos(-1/3) = 1.91063

The area under the curve y = ex from x = 0 to x = k is 1. Find the value of k.

k = ln 2

Find the value(s) of x to the nearest hundredth at which the slope of the line tangent to the graph of y = 2 ln(x2 + 3) is equal to - 1/2 .

x = -7.61, x = -0.39

If an open box is to be made using a square sheet of tin, 20 inches by 20 inches, by cutting a square from each corner and folding the sides up, find the length of a side of the square being cut so that the box will have a maximum volume.

x = 10/3

Using your calculator, find the value(s) of x at which the graphs of y = 2x2 and y = ex have parallel tangents.

x = 2.15 and x = 0.36

If for 0 ≤ x ≤ 2π, find the values(s) of x where f has a local minimum.

x = 3π/2

Write an equation of the line tangent to the graph of y = -3 sin(2x) at x = π/2.

y = 6x - 3π

Find the volume of the solid generated by revolving about the x -axis the region bounded by the graph of , where 0 ≤ x ≤ π/2 , the x -axis, and the y- axis.

π

When the area of a square is increasing twice as fast as its diagonals, what is the length of a side of the square?

√2 units