Calculus 1 Quiz: Accumulation Function Behavior

Let $f(t)$ be a continuous function. Define $G(x) = \int_{-x}^{x^2} f(t) dt$. Which of the following is an expression for $G'(x)$?

Let $f(t)$ be a continuous function and define $G(x) = \int_{1}^{x} f(t) \,dt$. If the average value of $f$ on the interval $[1, 5]$ is 10, what is the value of $G(5)$?

Let $G(x) = \int_{0}^{x} f(t) \,dt$. If $f$ is a continuous and strictly decreasing function on $[0, 5]$ and $f(2)=0$, which of the following statements must be true?

Let $f(t)$ be a continuous and strictly positive function for all real numbers $t$. Let $H(x) = \int_{0}^{1-x} f(t) \,dt$. Which statement accurately describes the function $H(x)$?

Let $G(x) = \int_{a}^{x} f(t) \,dt$, where $f$ is a continuous function. If $G(x)$ is always concave up, which of the following must be true about the function $f(t)$?

Let $f(x)$ be a differentiable function with $f(1)=4$. Let $G(x) = x^2 \int_1^x f(t) dt$. Find $G'(1)$.

Let $f(t)$ be a differentiable function. The function $f$ is decreasing on $(-\infty, -1)$, increasing on $(-1, 4)$, and decreasing on $(4, \infty)$. Let $G(x) = \int_{0}^{x} f(t) \,dt$. Which of the following statements describes the concavity of the graph of $G(x)$?

Let $G(x) = \int_{-2}^{x} f(t) \,dt$ and $H(x) = \int_{-2}^{x} |f(t)| \,dt$. Suppose $f(t)$ is a continuous function such that $f(t) > 0$ on $[-2, 0)$ and $f(t) < 0$ on $(0, 2]$. Which of the following statements is true for $x \in (0, 2]$?

The velocity of a particle is given by $v(t) = t^2 - 4t + 3$ for $t \ge 0$. Let $P(x) = \int_0^x v(t) dt$ be the particle's displacement. The total distance traveled by the particle on the interval $[0, 3]$ is given by $D = \int_0^3 |v(t)| dt$. Which of the following correctly relates $P(3)$ and $D$?

Let $f(t)$ be a continuous function such that $f(t) > 0$ on the interval $(1, 4)$ and $f(t) < 0$ on the intervals $(-\infty, 1)$ and $(4, \infty)$. Let $G(x) = \int_{2}^{x} f(t) \,dt$. At which value of $x$ does $G(x)$ have a local maximum?

Let $f(t)$ be a differentiable function that is increasing on the interval $(-\infty, 3)$ and decreasing on the interval $(3, \infty)$. Let $G(x) = \int_{-1}^{x} f(t) \,dt$. On which interval is the graph of $G(x)$ concave up?

Let $f(t)$ be a continuous function on $[-4, 6]$. The only zero of $f(t)$ in the interval $(-4, 6)$ is at $t = 3$. We know that $f(t) > 0$ for $t \in [-4, 3)$ and $f(t) < 0$ for $t \in (3, 6]$. Let $G(x) = \int_{-4}^{x} f(t) \,dt$. At what value of $x$ does $G(x)$ attain its absolute maximum value on the interval $[-4, 6]$?

Let $f(t)$ be a continuous function. The area of the region bounded by the graph of $f$, the t-axis, and the lines $t=0$ and $t=4$ is 10. The area of the region bounded by the graph of $f$, the t-axis, and the lines $t=4$ and $t=7$ is 3. Given that $f(t) > 0$ for $t \in [0,4]$ and $f(t) < 0$ for $t \in [4,7]$, what is the value of $\int_{0}^{7} f(t) \,dt$?

Let $f(t)$ be a continuous function and define $F(x) = \int_0^x f(t) dt$. Let $H(x) = \int_0^x F(t) dt$. If $f(t)$ is positive and increasing for $t \ge 0$, which of the following describes the graph of $H(x)$ for $x > 0$?

Let $f(t)$ be a continuous, positive, and increasing function on the interval $[0, \infty)$. Let $A = \int_{1}^{3} f(t) \,dt$ and $B = \int_{2}^{4} f(t) \,dt$. Which of the following best describes the relationship between $A$ and $B$?

Let $f(t)$ be a differentiable function with $f(3)=0$ and $f'(3) = -2$. Define $G(x) = \int_{0}^{x} f(t) \,dt$. What can be concluded about $G(x)$ at $x=3$?

Let $f(t)$ be a continuous function that is always positive. Let $G(x) = \int_{x}^{0} f(t) \,dt$. Which of the following must be true about $G(x)$ for $x > 0$?

Let $f(t)$ be a continuous function with $f(2) = 5$. Define $G(x) = \int_{1}^{x^2} f(t) \,dt$. What is the slope of the tangent line to the graph of $y=G(x)$ at $x=\sqrt{2}$?

Let $G(x) = \int_a^x f(t) dt$, where $f(t)$ is a linear function with a negative slope. Which of the following statements provides the best description of the graph of $G(x)$?

Let $f(t)$ be an odd, continuous function, meaning $f(-t) = -f(t)$ for all $t$. Let $G(x) = \int_{-2}^{x} f(t) \,dt$. If $G(2) = 10$, what is the value of $G(-2)$?