Let $$f$$ be a continuous function such that $$f(0) = 3$$, $$f(2) = -1$$, $$f(4) = 2$$, and $$f(6) = 0$$. If $$g(x) = \int_{0}^{x} f(t) dt$$, at what value of $$x$$ does $$g$$ attain its absolute maximum on the interval $$[0,6]$$?
Functions defined by integrals, their properties, behavior, and extrema.
41 questions
AP Learning Target(s): LT 4: The Fundamental Theorem of Calculus and Accumulation Functions, LT 5: Interpreting the Behavior of Accumulation Functions Involving Area
Let $$f$$ be a continuous function such that $$f(0) = 3$$, $$f(2) = -1$$, $$f(4) = 2$$, and $$f(6) = 0$$. If $$g(x) = \int_{0}^{x} f(t) dt$$, at what value of $$x$$ does $$g$$ attain its absolute maximum on the interval $$[0,6]$$?
x = 0
x = 6
x = 2
x = 4
The water level in a reservoir is modeled by the function $$h(t) = 0.5t^2 + 2t + 1$$, where $$h(t)$$ is in meters and $$t$$ is in hours. First, determine the total change in water level from $$t=1$$ to $$t=4$$ by integrating the rate of change of $$h(t)$$. Second, explain the significance of the constant of integration in this calculation.
15 and the constant of integration cannot be ignored in accumulation calculations.
12 and the constant of integration indicates the initial water level.
13.5 and the constant of integration represents an arbitrary constant that cancels in definite integrals.
13.5 but the constant of integration must be added to the result from the integral.
If a quantity's rate of change is negative over an interval, which of the following statements describes the accumulated change of the quantity over that same interval?
Accumulated change is infinite
Accumulated change is zero
Accumulated change is positive
Accumulated change is negative
If the rate of change of a function is positive over a given interval, which of the following must be true about the accumulated change over that interval?
Accumulated change is negative
Accumulated change is undefined
Accumulated change is positive
Accumulated change is zero
If the rate of change of a function is zero over an interval, what is the accumulated change of the function over that interval?
Accumulated change is zero
Accumulated change is undefined
Accumulated change is negative
Accumulated change is positive
The area under the curve of a function modeling a rate of change represents which of the following?
The accumulation of change
The instantaneous rate of change
The average rate of change
The maximum rate of change
The area of the region under the graph of a function over an interval represents which of the following?
The instantaneous rate of change
The maximum rate of change
The average rate of change
The accumulation of change
The function $$f$$ is defined by $$f(x) = \displaystyle \int_{1}^{x} \sqrt{t^3 + 2} dt$$. What is the average rate of change of $$f$$ on the interval $$[0, 3]$$?
1.324
1.497
2.266
2.694
If the graph of $f$, the derivative of an integrally-defined function $h$, is below the x-axis on the interval $(e, f)$, what can be concluded about the behavior of $h$ on that interval?
$h(x)$ is concave down on $(e, f)$.
$h(x)$ is decreasing on $(e, f)$.
$h(x)$ is increasing on $(e, f)$.
$h(x)$ is concave up on $(e, f)$.
Let `f` be the function defined by `f(x) = $$\int$$(0 to x) (t-1)/(t²+1) dt`. Which of the following statements is true?
f(x) has exactly one critical point
f(x) has no critical points
f(x) has infinitely many critical points
f(x) has exactly two critical points
Which of the following is represented by the definite integral of a function over an interval?
The rate of change of the function at a specific point
The maximum value of the function over a given interval
The average value of the function over a given interval
The accumulation of change in the function over a given interval
The rate at which water flows into a tank, in gallons per minute, is given by r(t) = 3t² - 2t + 5. If F(x) represents the total amount of water that has flowed into the tank over the time interval from t = 0 to t = x, which of the following is an expression for F(x)?
F(x) = x³ - x² + 5x
F(x) = $\displaystyle\int_{0}$ˣ (6t - 2) dt
F(x) = 3x² - 2x + 5
F(x) = $\displaystyle\int_{0}$ˣ (3t² - 2t + 5) dt
Which of the following statements best defines an integrally-defined function?
A function defined using a definite integral.
A function defined using a power series.
A function defined using an indefinite integral.
A function defined using a derivative.
Let `f` be the function defined by `f(x) = $\displaystyle\int_{0}$ˣ e^(-t) dt`. Which of the following statements is true?
lim(x→∞) f(x) = e
f(x) is unbounded as x approaches infinity
f(x) approaches 0 as x approaches infinity
lim(x→∞) f(x) = 1
The total accumulated volume of water in a tank over a one-hour period is given by the definite integral $$\int_{0}^{1}(5+2t) dt$$. Which of the following correctly evaluates this integral?
Total Volume = $$5*1+2*1=7$$, by multiplying each term by the time interval.
Total Volume = $$[5*t+t^2]_{0}^{1}=(5+1)-0=6$$
Total Volume = $$\int_{0}^{1}5 dt+2=5+2=7$$, incorrectly combining integration with an additional constant.
Total Volume = $$5+2=7$$, by simply adding the constants.
If $f$ is a continuous function such that $$\int_{1}^{x} f(t) dt = x^3 - 3x + 2$$, what is the value of $f(4)$?
3x^2 - 3
45
52
38
The function F is defined by F(x) = $\displaystyle\int_{1}$ˣ (ln t)/t dt. What is the value of F(e²)?
2
ln(e²)
e²
4
Let $f$ be the derivative of an integrally-defined function $h$. If $f(x)$ changes from negative to positive at $x = k$, what can be concluded about the function $h$ at $x = k$?
$h(x)$ has a relative minimum at $x = k$.
$h(x)$ is concave down at $x = k$.
$h(x)$ is concave up at $x = k$.
$h(x)$ has a relative maximum at $x = k$.
If the graph of $f(x)$, the derivative of an integrally-defined function $h(x)$, changes from positive to negative at $x = l$, what can be concluded about $h(x)$ at $x = l$?
$h(x)$ has a relative maximum at $x = l$.
$h(x)$ has a relative minimum at $x = l$.
$h(x)$ is concave down at $x = l$.
$h(x)$ is concave up at $x = l$.
$$If `F(x) = \int_{2}^{x} (3t^2 - 4) dt`, then `F(3) =`$$
15
11
19
7
The function f is defined by f(x) = $\displaystyle\int_{0}$ˣ e^(-t) dt. Which of the following is equivalent to f(x)?
f(x) = e^x - 1
f(x) = 1 - e^(-x)
f(x) = -ln(e^(-x))
f(x) = e^(-x)
Given the functions f(x) = sin(x²) and F(x) = $\displaystyle\int_{0}$ˣ sin(t²) dt, which of the following statements is true?
F(x) = -cos(x²)
F'(x) = sin(x²)
F(x) = $\displaystyle\int_{0}$ˣ sin(t²) dt = F(0) - F(x)
F(x) cannot be expressed in terms of elementary functions
The graph of the function $$G(x) = \int_{0}^{x} g(t)dt$$ has a point of inflection under which of the following conditions for the function $$g$$?
When g(t) reaches its maximum or minimum value
When the integral of g(t) equals zero
When g(t) has a discontinuity
When g(t) changes from positive to negative or vice versa
Which of the following represents the accumulation of change for a function `f` on the interval `[a, b]`?
$\displaystyle\int_{a}^{b}$ f'(x) dx
f'(b) - f'(a)
$\displaystyle\int_{a}^{b}$ f(x) dx
f(b) - f(a)
What does the definite integral of a negative function represent?
The negative rate of change
The maximum rate of change
The average rate of change
The accumulation of change
For a continuous function representing a rate of change, how is the total accumulation of change over an interval determined?
By solving a differential equation
By evaluating a definite integral
By calculating the limit
By finding the derivative
Let the function $$h$$ be defined by $$h(x) = \int_{1}^{x} (t^3 - 4t) dt$$. At what value(s) of $$x$$ in the interval $$[1, 3]$$ does $$h$$ have a local maximum?
$$x = \sqrt{4/3}$$
There are no local maxima in [1, 3]
$$x = 2$$
$$x = 1 and x = 3$$
Let g(x) = $\displaystyle\int_{1}$ˣ f(t) dt, where f is a continuous function such that f(t) > 0 for 1 < t < 3 and f(t) < 0 for 3 < t < 7. At which value of x does g attain its maximum value?
x = 1
x = 5
x = 7
x = 3
$$Which of the following best describes the quantity represented by the accumulation function `F(x) = \int_{a}^{x} f(t) dt`?$$
It measures instantaneous rate at `x`
It measures total accumulation from `a` to `x`
It measures total change from `0` to `x`
It measures average rate of change from `a` to `x`
When using a definite integral to calculate the total accumulation of a quantity, which of the following presents the correct sequence of steps?
Solve the integral and interpret the result, Define the variable and function representing rate, Determine the limits of integration based on the context, Set up the definite integral modeling the accumulation
Determine the limits of integration based on the context, Define the variable and function representing rate, Solve the integral and interpret the result, Set up the definite integral modeling the accumulation
Define the variable and function representing rate, Determine the limits of integration based on the context, Set up the definite integral modeling the accumulation, Solve the integral and interpret the result
Set up the definite integral modeling the accumulation, Define the variable and function representing rate, Determine the limits of integration based on the context, Solve the integral and interpret the result
Let `f` be a continuous function on the interval `[a, b]`. Suppose `f(x) > 0` for all `x` in `[a, b]` except at a single point `c`, where `f(c) = 0`. If `G(x) = $$\int$$ₐˣ f(t) dt`, which of the following must be true about `G`?
G'(c) = 0
G(x) has a local maximum at x = c
G(x) is constant at x = c
G(x) is strictly increasing on [a, b]
Let `f` be a continuous function where `f(x) > 0` for `x < 2` and `f(x) < 0` for `x > 2`. If the function `g` is defined by `g(x) = $\displaystyle\int_{0}$ˣ f(t) dt` and `g(4) = -3`, which of the following statements must be true?
g(x) is always decreasing on [0,4]
g(2) > 0
g(1) < 0
g(3) > g(2)
$$Let the function `F` be defined as `F(x) = \int_{a}^{x} f(t) dt`. If the function `f` is always positive, which of the following must be true about `F`?$$
F(x) is constant
F(x) is decreasing
F(x) is increasing
F(x) is oscillating
The function *h* is defined by h(x) = $\displaystyle\int_{1}$ˣ cos(πt) dt. Which of the following is equivalent to h(3) - h(2)?
(1/π)[sin(3π) - sin(2π)]
$\displaystyle\int_{1}^{3}$ cos(πt) dt - $\displaystyle\int_{1}^{2}$ cos(πt) dt
$\displaystyle\int_{2}^{3}$ cos(πt) dt
cos(3π) - cos(2π)
Let `f` be the function defined by `f(x) = $\displaystyle\int_{0}$ˣ e^(-t) dt`. Which of the following expressions is equivalent to `f(a+h) - f(a)`?
h·e^(-a)
$\displaystyle\int_{0}$ᵃ⁺ʰ e^(-t) dt - $\displaystyle\int_{0}$ᵃ e^(-t) dt
$$\int$$ₐᵃ⁺ʰ e^(-t) dt
e^(-a-h) - e^(-a)
The function $$h$$ is defined by $$h(x) = \int_{1}^{x} (t^2 - 4t + 3) dt$$. For which value of $$x$$ other than $$x=1$$ is $$h(x)=0$$?
x = 2
x = 3
x = 4
x = 1
Let the function F be defined by F(x) = $\displaystyle\int_{1}$ˣ cos(πt) dt. What is the equation of the line tangent to the graph of F at x = 3/2?
y = -1(x - 3/2) + F(3/2)
y = 0(x - 3/2) + F(3/2)
y = cos(3π/2)(x - 3/2) + F(3/2)
y = π·sin(3π/2)(x - 3/2) + F(3/2)
What is an equation of the line tangent to the graph of $$y = 3 - \displaystyle\int_{-1}^{x} e^{-t^3} dt$$ at $$x = -1$$?
$$y - 3 = -3e(x + 1)$$
$$y - 3 = -e(x + 1)$$
$$y - 3 = 0$$
$$y - 3 = \displaystyle \frac{1}{e}(x + 1)$$
Let the function h be defined by h(x) = $\displaystyle\int_{2}$ˣ ln(t)/t dt. What is the slope of the line tangent to the graph of h at x = e?
ln(e)
ln(e)/e
1/e
1
Let g be the function defined by g(x) = $\displaystyle\int_{0}$ˣ √(1 + t³) dt. The tangent line to the graph of g at x = 1 passes through the point (3, k). What is the value of k?
2√2 + g(1)
√2 + g(1)
2√2
2 + g(1)
The rate of change of a population P (in thousands) with respect to time t (in decades) is given by dP/dt = 0.03P. What are the units of $\displaystyle\int_{1}^{3}$ 0.03P dt?
0.03 × thousands
thousands per decade
thousands
thousands × decades