Master Calculus Applications: Enhance Your Problem-Solving Skills

Question 1

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. At the moment when the balloon's radius is 5 centimeters, what is the rate of change of its radius?

Accepted Answer

0.2 cm/s

Answer

0.5 cm/s

Answer

1 cm/s

Answer

2 cm/s

Question 2

The velocity of a car, expressed as a function of time, is v(t) = 2t - 5. Calculate the car's acceleration.

Accepted Answer

2 feet per second squared

Answer

1 foot per second squared

Answer

5 feet per second squared

Answer

10 feet per second squared

Question 3

Determine the derivative of the function g(x) = e^x + ln(x).

Accepted Answer

e^x + 1/x

Answer

e^x - 1/x

Answer

e^x + x

Answer

e^x - x

Question 4

Find the vertex of the graph of the function f(x) = x^2 - 4x + 3.

Accepted Answer

(2, -1)

Answer

(1, 0)

Answer

(3, 2)

Answer

(4, 5)

Question 5

A particle's position is described by the function s(t) = t³ - 2t² + 3t. What is its acceleration at time t = 2 seconds?

Accepted Answer

6

Answer

0

Answer

3

Answer

12

Question 6

What is the derivative of the function y = ln(x² + 1)?

Accepted Answer

2x / (x² + 1)

Answer

1 / (x² + 1)

Answer

-1 / (x² + 1)

Answer

2x

Question 7

Which scenario exemplifies a related rate problem?

Accepted Answer

Determining the velocity of a moving vehicle

Answer

Calculating the area of a triangle

Answer

Determining the volume of a sphere

Answer

Finding the midpoint of a line segment

Question 8

Which integral represents the volume of the solid generated when the region bounded by y = x² and y = 4 is rotated about the x-axis?

Accepted Answer

∫[0, 2] π(4 - x²) dx

Answer

∫[0, 2] 2πx(4 - x²) dx

Answer

∫[0, 2] π(x² - 4) dx

Answer

∫[0, 2] 2πx(x² - 4) dx

Question 9

A projectile is launched vertically upward with an initial velocity of 100 m/s. What is its maximum height?

Accepted Answer

122.5 m

Answer

245 m

Answer

490 m

Answer

980 m

Question 10

A car is traveling at a constant speed of 60 mph. What is its acceleration?

Accepted Answer

0 m/s^2

Answer

17.9 m/s^2

Answer

35.8 m/s^2

Answer

71.6 m/s^2

Question 11

A population of bacteria grows exponentially at a rate of 5% per hour. If the initial population is 1000, what will be the population after 10 hours?

Accepted Answer

1644

Answer

1280

Answer

1500

Answer

2000

Question 12

Find the antiderivative of the function f(x) = e^x.

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

1/e^x + C

Answer

ln(e^x) + C

Question 13

Which of these is NOT an application of calculus in physics?

Accepted Answer

Calculating the area of a region under a curve

Answer

Calculating the trajectory of a projectile

Answer

Determining the acceleration of an object in motion

Answer

Solving differential equations

Question 14

A spherical balloon is being inflated at a rate of 10 cm³ per second. Find the rate of change of the balloon's radius when the radius is 5 cm.

Accepted Answer

1.25 cm/s

Answer

1 cm/s

Answer

2 cm/s

Answer

2.5 cm/s

Question 15

Find the derivative of the function f(x) = ∫₀ˣ (t³ - t) dt.

Accepted Answer

x³ - x

Answer

x³ - 1

Answer

x² - 1

Answer

x² - x

Question 16

A company's revenue function is R(x) = 100x - x^2, where x is the number of units sold. What is the change in revenue when one additional unit is sold when 4 units have already been sold?

Accepted Answer

80

Answer

100

Answer

40

Answer

60

Question 17

A bacteria colony grows exponentially at a rate of 5% per hour. If the initial population is 1000, what will be the population after 10 hours?

Accepted Answer

1628

Answer

1050

Answer

2500

Answer

10000

Question 18

A car's speed is given by the function v(t) = 3t^2 - 2t + 1, where t is time in seconds. At what time does the car reach its highest speed?

Accepted Answer

t = 1/3

Answer

t = 0

Answer

t = 1

Answer

t = 2

Question 19

Water is flowing into a cylindrical tank at a rate of 5 cubic meters per minute. The tank has a radius of 2 meters. What is the rate of change of the water level (in meters per minute) when the height of the water is 3 meters?

Accepted Answer

0.625

Answer

1

Answer

0.75

Answer

0.5

Question 20

A company's profit function is P(x) = -x^3 + 12x^2 - 45x + 50, where x represents the number of units sold. How many units should be sold to maximize profit?

Accepted Answer

8

Answer

4

Answer

16

Answer

12

Question 21

A rectangular garden has an area of 100 square meters. What are the dimensions (length and width) that minimize the amount of fencing needed to enclose the garden?

Accepted Answer

Width: 10 meters, Length: 10 meters

Answer

Width: 5 meters, Length: 20 meters

Answer

Width: 4 meters, Length: 25 meters

Question 22

The position of a particle at time *t* is given by the function  *s(t) = t^3 - 3t^2 + 2t*.  What is the velocity of the particle at *t = 2*?

Accepted Answer

2 m/s

Answer

8 m/s

Answer

6 m/s

Answer

4 m/s

Question 23

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. How fast is the surface area of the balloon increasing when the radius is 5 cm?

Accepted Answer

20π cm^2/s

Answer

10π cm^2/s

Answer

40π cm^2/s

Answer

30π cm^2/s

Question 24

The demand function for a product is given by *p(x) = 100 - 0.5x*, where *x* is the number of units sold and *p* is the price. If the fixed costs are $100 and the variable costs are $20 per unit, how many units need to be sold to break even?

Accepted Answer

400 units

Answer

800 units

Answer

200 units

Answer

600 units

Question 25

A projectile is launched straight upward with a starting speed of 40 meters per second. Ignoring air resistance, what is the highest point the projectile will reach?

Accepted Answer

80 meters

Answer

40 meters

Answer

160 meters

Answer

20 meters

Question 26

A bacteria population grows according to the logistic equation: dN/dt = 0.5N(1 - N/1000). What is the maximum population size that this environment can support?

Accepted Answer

1000

Answer

500

Answer

Unlimited

Answer

2000

Question 27

We want to find the area under the curve y = e^x from x = 0 to x = 1. If we use the trapezoidal rule with 4 equal sections, what is the approximate value of the area?

Accepted Answer

1.72

Answer

2.72

Answer

0.86

Answer

1.36

Question 28

A ball is thrown straight up with an initial velocity of 30 m/s. Its height (h) after time (t) seconds is given by the equation h(t) = 30t - 4.9t^2. What is the ball's maximum height?

Accepted Answer

45.92 m

Answer

30 m

Answer

15 m

Answer

60 m

Question 29

The marginal cost (MC) for producing x units of a product is given by MC(x) = 2x + 5.  If the fixed cost is $10, what is the total cost function (C(x))?

Accepted Answer

C(x) = x^2 + 5x + 10

Answer

C(x) = x^2 + 10

Answer

C(x) = 2x^2 + 5x + 10

Answer

C(x) = 2x + 15

Question 30

The rate of change of a population is directly proportional to the current population size. If the population doubles every 10 years, what is the continuous growth rate?

Accepted Answer

ln(2)/10

Answer

10/ln(2)

Answer

2

Answer

0.1

Question 31

The velocity of a particle moving along a line is given by v(t) = t^2 - 4t + 3. What is the particle's total displacement from time t = 0 to t = 3?

Accepted Answer

0

Answer

9

Answer

-3

Answer

3

Question 32

A company's total revenue (R) from selling x units of a product is given by R(x) = 100x - x^2. What is the marginal revenue function (MR(x)), which represents the additional revenue from selling one more unit?

Accepted Answer

MR(x) = 100 - 2x

Answer

MR(x) = -2x

Answer

MR(x) = 100x - x^2

Answer

MR(x) = 100 - x

Question 33

A farmer wants to enclose a rectangular field next to a river using 1000 meters of fencing.  The side along the river doesn't need a fence. What are the dimensions of the rectangle that will maximize the enclosed area?

Accepted Answer

250 meters by 500 meters

Answer

500 meters by 500 meters

Answer

200 meters by 600 meters

Answer

100 meters by 800 meters

Question 34

Imagine rotating the area under the curve y = sqrt(x) from x = 0 to x = 4 around the x-axis. This creates a 3D shape. What is the volume of this shape?

Accepted Answer

32π/3

Answer

64π/3

Answer

8π/3

Answer

16π

Question 35

A particle moves along a straight line with its velocity given by v(t) = 3t^2 - 6t. What is the particle's acceleration at t = 2?

Accepted Answer

6

Answer

12

Answer

0

Answer

-6

Question 36

A particle moves in a straight line with a velocity given by v(t) = t^2 - 2t + 3. What is the total distance traveled by the particle in the first 3 seconds?

Accepted Answer

8.5

Answer

12.5

Answer

16.5

Answer

4.5

Question 37

A ball is thrown vertically upward with an initial velocity of 20 meters per second. How long will it take the ball to reach its maximum height?

Accepted Answer

5

Answer

2.5

Answer

10

Answer

7.5

Question 38

A ball is thrown vertically upward. Its height (in feet) after t seconds is given by the function h(t) = 80t - 16t². What is the maximum height the ball reaches?

Accepted Answer

100 feet

Answer

160 feet

Answer

80 feet

Answer

50 feet

Question 39

The rate of decay of a radioactive substance is proportional to the amount present. If 200 grams of the substance decays to 50 grams in 10 years, what is the half-life of the substance?

Accepted Answer

5 years

Answer

10 years

Answer

2.5 years

Answer

20 years

Question 40

A company's marginal cost function is given by MC(x) = 3x² + 2x + 5, where x is the number of units produced. Find the total cost of producing the first 10 units.

Accepted Answer

$1150

Answer

$370

Answer

$1100

Answer

$325

Question 41

The area bounded by the curve y = x², the x-axis, and the lines x = 1 and x = 3 is:

Accepted Answer

26/3 square units

Answer

8 square units

Answer

9 square units

Answer

28/3 square units

Question 42

A cylindrical tank with radius 5 meters is being filled with water at a rate of 3 cubic meters per minute. At what rate is the height of the water level rising?

Accepted Answer

3/(25π) meters per minute

Answer

75π meters per minute

Answer

15π meters per minute

Answer

3π/25 meters per minute

Question 43

A rectangular box with a square base is to be constructed from 1200 square inches of material. What is the maximum possible volume of the box?

Accepted Answer

4000 cubic inches

Answer

2500 cubic inches

Answer

2000 cubic inches

Answer

3000 cubic inches

Question 44

The population of a city is growing at a rate proportional to its size. If the population doubles in 20 years, what is the growth constant (k) for the population?

Accepted Answer

ln(2)/20

Answer

2

Answer

ln(2)

Answer

20

Question 45

A company's profit function is given by P(x) = -0.02x² + 40x - 5000, where x is the number of units sold. How many units should be sold to maximize profit?

Accepted Answer

1000 units

Answer

1500 units

Answer

2000 units

Answer

500 units

Question 46

Find the average value of the function f(x) = x³ - 6x on the interval [-2, 2].

Accepted Answer

-4

Answer

-8

Answer

8

Answer

0

Question 47

A company's profit function is given by P(x) = -x² + 100x - 1000, where x is the number of units produced. What is the maximum profit the company can achieve?

Accepted Answer

2500

Answer

1000

Answer

100

Answer

5000

Question 48

The marginal cost of producing x units of a product is given by MC(x) = 2x + 5. If the fixed cost is $100, what is the total cost function C(x)?

Accepted Answer

C(x) = x² + 5x + 100

Answer

C(x) = 2x² + 5x

Answer

C(x) = x² + 100

Question 49

The rate of change of a population of bacteria is given by dP/dt = 0.05P, where P is the population size and t is time in hours. If the initial population is 1000, what is the population after 2 hours?

Accepted Answer

1105

Answer

1050

Answer

1000

Answer

1200

Question 50

The area under the curve of f(x) = x² from x = 0 to x = 2 is represented by which definite integral?

Accepted Answer

∫₀² x² dx

Answer

∫₀² x dx

Answer

∫₀² 2x dx

Answer

∫₀² 2 dx

Question 51

The velocity of a particle is given by v(t) = 3t² - 4t. What is the displacement of the particle from t = 1 to t = 3?

Accepted Answer

10

Answer

8

Answer

6

Answer

12

Question 52

A company's revenue function is R(x) = 100x - 0.5x² and its cost function is C(x) = 20x + 100. What is the marginal profit when x = 50?

Accepted Answer

50

Answer

100

Answer

0

Answer

-50

Question 53

The volume of a sphere is given by V(r) = (4/3)πr³. What is the rate of change of the volume with respect to the radius when r = 2?

Accepted Answer

16π

Answer

4π

Answer

8π

Answer

32π

Question 54

The demand function for a product is given by p = 100 - 2q, where p is the price and q is the quantity. What is the consumer surplus when the price is $60?

Accepted Answer

400

Answer

100

Answer

800

Answer

200

Question 55

A function f(x) is defined as f(x) = ∫₀ˣ (t² + 1) dt. What is the value of f'(2)?

Accepted Answer

5

Answer

2

Answer

1

Answer

4