Calculus 2 Quiz: Accumulation Function Behavior

Let $f(t)$ be a continuous, positive, and decreasing function for all $t \ge 0$. Let $A(x) = \int_0^x f(t) \, dt$. Which of the following inequalities must be true for $x > 0$?

The function $f(t)$ is continuous and has a single root at $t=c$. The accumulation function $G(x) = \int_a^x f(t) \, dt$ is decreasing for $x<c$ and increasing for $x>c$. Which of the following must be true?

Let $f(t)$ be a continuous function that is positive on the interval $(-\infty, 3)$ and negative on the interval $(3, \infty)$. Define an accumulation function $G(x) = \int_1^x f(t) \, dt$. Which of the following statements correctly describes $G(x)$ at $x=3$?

Let $f(t)$ be a continuous function such that $f(t) < 0$ on $(-\infty, 2)$ and $f(t) > 0$ on $(2, \infty)$, with $f(2)=0$. Define $F(x) = \int_4^x f(t) \, dt$. Which of the following statements about the value of the local extremum of $F(x)$ is true?

Let $f(t)$ be a differentiable function that is decreasing on the interval $(-\infty, 0)$ and increasing on the interval $(0, \infty)$. It is also known that $f(0) < 0$. Let $H(x) = \int_5^x f(t) \, dt$. Which statement accurately describes the behavior of $H(x)$ at $x=0$?

Let $f(t)$ be a twice-differentiable function. The function $f(t)$ has a local maximum at $t=4$ and a single root at $t=1$. Let $F(x) = \int_0^x f(t) \, dt$. On what interval must the graph of $F(x)$ be concave down?

Let $f(t)$ be an odd continuous function, i.e., $f(-t) = -f(t)$ for all $t$. Let $F(x) = \int_{-3}^x f(t) \, dt$. If $\int_0^3 f(t) \, dt = 5$, what is the value of $F(3)$?

Let $f(t)$ be a continuous function. Define $G(x) = \int_0^x f(t) \, dt$. If $G(x)$ is always increasing for $x>0$ and is concave down for $x>0$, which of the following must be true about $f(t)$ for $t>0$?

Let $f(t)$ be a continuous function such that $f(t) < 0$ for all $t$. Define $F(x) = \int_1^{x^2} f(t) \, dt$. For what values of $x$ is $F(x)$ increasing?

Let $F(x) = \int_x^{x+3} t^2 \, dt$. At what value of $x$ is the instantaneous rate of change of $F(x)$ equal to 21?

Let $f(t)$ be a continuous function, and define $F(x) = \int_0^x f(t) \, dt$. If $F(x)$ has a point of inflection at $x=3$ and a critical point at $x=5$, what can be concluded about the function $f$?

Let $f(t)$ be a continuous function and $F(x) = \int_{-1}^x f(t) \, dt$. Given that $F(-1)=0$ and $F(3)=0$, and $F(x) > 0$ on $(-1, 3)$, which of the following statements must be true about the function $f$?

Let $f(t)$ be a continuous function where $f(t) > 0$ on $(1, 3)$ and $f(t) < 0$ on $(0, 1) \cup (3, 4)$. It is known that $\int_1^3 f(t) \, dt = 6$ and $\int_3^4 f(t) \, dt = -6$. Define $G(x) = \int_1^x f(t) \, dt$. On the interval $[1, 4]$, the absolute maximum value of $G(x)$ occurs at which value of $x$?

Let $f(t)$ be a continuous function. Define $G(x) = \int_{-2}^x f(t) \, dt$. It is known that $f(t)$ is decreasing on $(-\infty, 3)$ and increasing on $(3, \infty)$. Which of the following statements must be true about $G(x)$?

Let $f(t)$ be a continuous function. Given $\int_0^3 f(t) \, dt = 6$, $\int_3^7 f(t) \, dt = -10$, and $\int_7^8 f(t) \, dt = 3$. Let $F(x) = \int_0^x f(t) \, dt$. Which of the following correctly orders the values of $F(0), F(3), F(7), F(8)$ from least to greatest?

Let $f(t)$ be a continuous function. The accumulation function $G(x) = \int_c^x f(t) \, dt$ has a local maximum at $x=a$ and a point of inflection at $x=b$. Which of the following conditions on $f$ must be met?

Let $f(t)$ and $g(t)$ be continuous positive functions for all $t \ge 0$, and suppose $f(t) > g(t)$ for all $t \ge 0$. Let $F(x) = \int_0^x f(t) \, dt$ and $G(x) = \int_0^x g(t) \, dt$. Which of the following statements is NOT necessarily true for $x > 0$?

Let $v(t)$ be a continuous velocity function for a particle moving along a line. The particle's velocity is positive on $(0, 2)$ and negative on $(2, 5)$. Let $D(T) = \int_0^T v(t) \, dt$ represent the particle's displacement at time $T$, and let $S(T) = \int_0^T |v(t)| \, dt$ represent the total distance traveled. Which of the following statements must be true for any time $T$ in the interval $(2, 5)$?

Let $f(t)$ be a continuous function such that $f(t) < 0$ on $(-\infty, -1)$ and $f(t) > 0$ on $(-1, \infty)$. Let the function $H(x)$ be defined as $H(x) = \int_x^2 f(t) \, dt$. Which of the following statements about $H(x)$ is correct?

Let $A(x) = \int_1^x |\sin(\pi t)| \, dt$. The function $A(x)$ represents the total area between $y = |\sin(\pi t)|$ and the $t$-axis from $t = 1$ to $t = x$. What is the behavior of $A'(x)$ at $x = 3$?