The concept of accumulation of change can be represented by the area under a curve. For a velocity-time graph, the area under the curve over a given time interval represents which of the following?
Using integrals for net change: position, velocity, displacement, distance, and rates.
33 questions
The concept of accumulation of change can be represented by the area under a curve. For a velocity-time graph, the area under the curve over a given time interval represents which of the following?
The area under the curve represents the average velocity over time
The area under the curve represents the total displacement over time
The area under the curve represents the instantaneous acceleration at each point
The area under the curve represents the total energy expended during motion
On a graph of velocity as a function of time, the area under the curve over an interval represents which of the following quantities?
The area under the curve represents the total displacement or distance traveled
The area under the curve represents the instantaneous velocity at any given point
The area under the curve represents the acceleration of the object over time
The area under the curve represents the change in velocity between two time points
The area under a velocity-time graph represents which of the following?
Instantaneous velocity
Average velocity
Total displacement
Total distance
The acceleration of a car is given by a(t) = 2 - 0.5t m/s² for 0 ≤ t ≤ 8. If the car starts from rest, what does the area under the graph of a from t = 0 to t = 4 represent?
The car's velocity at t = 4 seconds
The car's position at t = 4 seconds
The total distance traveled by the car from t = 0 to t = 4
The average acceleration from t = 0 to t = 4
The velocity of a particle moving along a line is given by `v(t) = 3t² - 6t + 2` meters per second for `0 ≤ t ≤ 4`. What is the physical interpretation of the area under the graph of `v(t)` from `t = 1` to `t = 3`?
The acceleration of the particle at t = 2
The total displacement of the particle from t = 1 to t = 3
The total distance traveled by the particle from t = 1 to t = 3
The average velocity of the particle from t = 1 to t = 3
Which of the following is an example of correctly calculating the displacement of a particle using the definite integral of the velocity function $$v(t)=3*t^2$$ for $$0\le t\le 2$$?
Displacement = $$3*(2-0)=6$$, by simply multiplying the constant value 3 by the time interval.
Displacement = $$3*2^2=12$$, computed by evaluating the function at the upper limit only.
Displacement = $$\frac{3*2^3}{3}=8$$, which arrives at 8 but via unclear reasoning.
Displacement = $$2^3-0^3=8$$, since the antiderivative is $$t^3$$.
Let `f` be a differentiable function. Which of the following is equivalent to the definite integral of its derivative, `f'`, over the interval `[a, b]`?
The definite integral is equal to the function evaluated at b minus the function evaluated at a.
The definite integral is equal to the average rate of change of the function over the interval.
The definite integral is equal to the derivative of the function evaluated at b minus the derivative of the function evaluated at a.
The definite integral is equal to the maximum rate of change of the function over the interval.
The definite integral of a function modeling the rate of change of a quantity represents which of the following?
Average rate of change
Slope of the tangent line
Accumulation of change
Instantaneous rate of change
The velocity of a runner is given by the function $$v(t) = 3t^2 - 2t$$, where $$v(t)$$ is measured in meters per second and $$t$$ is measured in seconds. What is the total distance the runner travels from time $$t=1$$ second to $$t=4$$ seconds?
50
45
48
52
What is the economic interpretation of the definite integral of a marginal cost function, `C'(x)`, from `x=a` to `x=b`?
Average cost
Fixed cost
Marginal revenue
Total cost change
The rate at which motor oil is leaking from an automobile is modeled by the function $$L(t) = 1 + \sin(t^2)$$, where $$L(t)$$ is measured in liters per hour and $$t$$ is measured in hours for $$t \geq 0$$. How much oil, in liters, leaks out of the automobile during the first half hour?
1.998 liters
0.541 liters
1.247 liters
0.969 liters
The rate at which motor oil leaks from an automobile, in liters per hour, is modeled by the function $$L(t) = 1 + \sin\left(t^2\right)$$, where $$t$$ is the time in hours for $$t \geq 0$$. What is the total amount of oil that leaks from the automobile during the first half hour?
$$0.969$$ liters
$$0.541$$ liters
$$1.998$$ liters
$$0.531$$ liters
The rate at which the depth of water in a rain gauge increases is modeled by the function $$R(t) = 0.5 + t \cos\left(\frac{\pi t^3}{80}\right)$$, where $$R(t)$$ is measured in centimeters per hour and $$t$$ is the time in hours since the start of a rainfall. What is the total increase in the depth of the water from time $$t = 0$$ to $$t = 3$$ hours?
1.966 cm
1.466 cm
1.233 cm
5.401 cm
The function $$r(t) = 10 - 2\sin\left(\frac{\pi t}{6}\right)$$ gives the rate, in gallons per minute, at which water flows into a tank, where $$t$$ is the time in minutes. What is the total amount of water that flows into the tank during the first 12 minutes?
$$120 - 24/\pi gallons$$
120 gallons
96 gallons
$$120 + 24/\pi gallons$$
The cost in dollars for a company to produce x units is modeled by the function $$C(x) = x^3 - 6x^2 + 9x + 15$$. What is the total change in cost as production increases from 2 to 5 units, calculated by the definite integral of the marginal cost function, $$C'(x)$$, over the interval $$[2, 5]$$?
22
20
18
15
The area under the graph of a constant function over an interval represents which of the following?
The average rate of change
The maximum rate of change
The accumulation of change
The instantaneous rate of change
Which of the following is represented by the definite integral of a positive function?
The average rate of change
The instantaneous rate of change
The accumulation of change
The maximum rate of change
All of the following statements about the interpretation of a definite integral as area are true EXCEPT:
The definite integral computes the net signed area, so parts of the curve below the x-axis contribute negatively.
To obtain the total geometric area irrespective of sign, one should integrate the absolute value of the function.
The definite integral always represents the total non-negative area between the function and the x-axis even if the function is negative over some intervals.
A negative value of a definite integral over an interval indicates that the function lies below the x-axis for that interval.
I.
$$\int_{0}^{4} f(x) dx$$ represents the net (algebraic) area between the curve and the x-axis.
II.
$$\left|\int_{0}^{4} f(x) dx\right|$$ represents the total area under the curve regardless of whether parts are below the x-axis.
III.
If $$f(x) \ge 0$$ on [0,4], then $$\int_{0}^{4} f(x) dx$$ equals the area of the region under the curve.
Which of these statements are true regarding the interpretation of definite integrals as areas?
I and III only
II only
I, II, and III
I only
The function `f` gives the rate of water flow into a tank. In this context, what does the integral of `f(t)` from `t=a` to `t=b` represent?
Total water added
Maximum flow rate
Average flow rate
Initial water volume
The rate of change of a population P(t) is modeled by the equation dP/dt = 200e^(-0.1t) - 50, where t is measured in years. What is the significance of the definite integral $\displaystyle\int_{0}^{4}$ (dP/dt) dt being negative?
The population decreased over the 4-year period
The population growth rate was decreasing over time
The population was negative at t = 4
The population will eventually become extinct
The marginal cost for a company to produce *x* units is given by *C'*(x) = 3x² - 12x + 15 dollars per unit. What is represented by the area of the region between the graph of *C'*(x) and the *x*-axis from *x* = 1 to *x* = 3?
The profit generated by producing 3 units instead of 1 unit
The minimum cost of production in the range 1 ≤ x ≤ 3
The average cost per unit when producing 3 units
The change in total cost when production increases from 1 to 3 units
The velocity of a car is given by the function $$v(t) = t - 3$$, measured in meters per second. Which statement accurately describes the car's net displacement from time $$t=0$$ to $$t=5$$ seconds?
The net displacement is $$7.5$$ m by incorrectly summing absolute values.
The net displacement is $$2.5$$ m, as the sign is ignored.
The net displacement is $$0$$ m by mistakenly assuming that positive and negative areas cancel completely.
The net displacement is $$-2.5$$ m.
All of the following statements about computing areas using definite integrals are true except:
The definite integral computes the net signed area, where areas below the x-axis are counted as negative.
The definite integral $$\int_{a}^{b} f(x) dx$$ always represents the total non-negative area between the curve and the x-axis.
To obtain the total geometric area without cancellations, one must integrate the absolute value of $$f(x)$$.
If $$f(x)$$ is negative over any part of the interval, then that portion subtracts from the integral's net value.
A particle moves along a line with velocity $$v(t) = 4t$$ m/s. What is the displacement of the particle from time $$t=1$$ to $$t=5$$?
The displacement is $$52$$ m because of an arithmetic error with the antiderivative.
The displacement is $$50$$ m due to a miscalculation of the time interval.
The displacement is $$24$$ m by incorrectly halving the result.
The displacement is $$48$$ m.
The velocity of a particle moving along the x-axis is given by v(t) = t² - 4t + 3 for 0 ≤ t ≤ 4. During which time interval is the particle moving to the left?
3 < t < 4
The particle never moves to the left
0 < t < 1
1 < t < 3
The velocity of a particle is given by v(t) = sin(t) - 2 for 0 ≤ t ≤ 2π. On which of the following intervals is the particle moving in the positive direction?
The particle never moves in the positive direction
π < t < 2π
0 < t < π
0 < t < 2π
The velocity of a particle is given by the function $v(t) = \sin(t)$ for $0 \le t \le 2\pi$. If the particle's initial position is $s(0) = 3$, which of the following correctly states the particle's position at $t=2\pi$ and provides the correct geometric justification?
3 + 2π, because the total area under v(t) is 2π
1, because sin(2π) = 0
3 - 2π, because the net signed area is negative
3, because the net area under v(t) from 0 to 2π is 0
Water flows into a tank at a rate modeled by the function r(t) = 5 - 2sin(πt/6), where r(t) is measured in gallons per minute and t is measured in minutes. For which time interval on 0 ≤ t ≤ 12 is the amount of water in the tank decreasing?
3 < t < 9
0 < t < 6
6 < t < 12
The amount of water never decreases
If the rate of change of a function is negative, which of the following must be true about the function?
Function is constant
Function is decreasing
Function is undefined
Function is increasing
If the rate of change of a function is always positive, which of the following statements about the function must be true?
Function is increasing
Function is constant
Function is oscillating
Function is decreasing
A particle moves with velocity $$v(t)=4*t$$. Evaluate the total distance traveled from t = 1 to t = 3.
$$20$$
$$12$$
$$8$$
$$16$$
The rate of sales for a new compact disc (CD) is modeled by the function $$R(t) = 20000 e^{-0.12 t}$$, where $$R(t)$$ is the number of CDs sold per week and $$t$$ is the number of weeks since the CD was released. To the nearest thousand, what is the total number of CDs sold during the first 12 weeks after the release?
11,000
5,000
127,000
57,000