ap calculus bc mcq practice test b 88. for all values of x, the continuous function f is positive and…

ap calculus bc mcq practice test b 88. for all values of x, the continuous function f is positive and decreasing. let g be the function given by g(x)=∫₂ˣf(t)dt. which of the following could be a table of values for g? (a) x g(x) 1 -2 2 0 3 1 (b) x g(x) 1 -2 2 0 3 3 (c) x g(x) 1 1 2 0 3 -2 (d) x g(x) 1 2 2 0 3 -1 (e) x g(x) 1 3 2 0 3 2

ap calculus bc mcq practice test b 88. for all values of x, the continuous function f is positive and decreasing. let g be the function given by g(x)=∫₂ˣf(t)dt. which of the following could be a table of values for g? (a) x g(x) 1 -2 2 0 3 1 (b) x g(x) 1 -2 2 0 3 3 (c) x g(x) 1 1 2 0 3 -2 (d) x g(x) 1 2 2 0 3 -1 (e) x g(x) 1 3 2 0 3 2

Answer

Explanation:

Step1: Recall the fundamental theorem of calculus

By the fundamental - theorem of calculus, $g^\prime(x)=f(x)$. Since $f(x)>0$ for all $x$, then $g^\prime(x)>0$ for all $x$, which means $g(x)$ is an increasing function. Also, $g(2)=\int_{2}^{2}f(t)dt = 0$.

Step2: Analyze each option

  • Option A: $g(1)= - 2$, $g(2)=0$, $g(3)=1$. But $g(1)<g(2)$ and $g$ should be increasing, so this is incorrect.
  • Option B: $g(1)= - 2$, $g(2)=0$, $g(3)=3$. Here, $g(1)<g(2)<g(3)$ and $g(2) = 0$, so this option is consistent with the properties of $g(x)$.
  • Option C: $g(1)=1$, $g(2)=0$, $g(3)= - 2$. Since $g(1)>g(2)$ and $g$ should be increasing, this is incorrect.
  • Option D: $g(1)=2$, $g(2)=0$, $g(3)= - 1$. Since $g(1)>g(2)$ and $g$ should be increasing, this is incorrect.
  • Option E: $g(1)=3$, $g(2)=0$, $g(3)=2$. Since $g(1)>g(2)$ and $g$ should be increasing, this is incorrect.

Answer:

B.