which of the following statements are true?\na) the fundamental theorem of calculus says if $f(t)$ is…

which of the following statements are true?\na) the fundamental theorem of calculus says if $f(t)$ is continuous, then $int_{a}^{b}f(t)dt = f(b)-f(a)$.\nb) if $c(q)$ is the cost to produce q units, then $int_{100}^{200}c(q)dq$ is the amount it costs to increase production from 100 to 200 units.\nboth a) and b)\nonly a)\nneither a) or b)\nonly b)\n10 multiple choice 1 point\ngiven the graph below, which of the following represents the shaded region in the graph?\n$f(t)=2t^{3}-2t + 3$\n$int_{-2}^{2}2t^{3}-2t + 3dt$\n$int_{-1}^{0.5}6t^{2}-2dt$\n$int_{-1}^{0.5}2t^{3}-2t + 3dt$\n$int_{-2}^{2}6t^{2}-2dt$

which of the following statements are true?\na) the fundamental theorem of calculus says if $f(t)$ is continuous, then $int_{a}^{b}f(t)dt = f(b)-f(a)$.\nb) if $c(q)$ is the cost to produce q units, then $int_{100}^{200}c(q)dq$ is the amount it costs to increase production from 100 to 200 units.\nboth a) and b)\nonly a)\nneither a) or b)\nonly b)\n10 multiple choice 1 point\ngiven the graph below, which of the following represents the shaded region in the graph?\n$f(t)=2t^{3}-2t + 3$\n$int_{-2}^{2}2t^{3}-2t + 3dt$\n$int_{-1}^{0.5}6t^{2}-2dt$\n$int_{-1}^{0.5}2t^{3}-2t + 3dt$\n$int_{-2}^{2}6t^{2}-2dt$

Answer

Explanation:

For the first question: "Which of the following statements are true?"

Step1: Analyze statement (a) regarding the Fundamental Theorem of Calculus.

Statement (a) misstates the FTC. Integrand should be $f(t)$, not $F(t)$. False.

Step2: Analyze statement (b) regarding cost increase.

Statement (b) correctly applies FTC. $\int C'(q)dq$ is change in cost. True. $$ \int_{100}^{200} C'(q) dq = C(200) - C(100) $$

Step3: Determine the true statement(s).

With (a) false and (b) true, only (b) is true.

For the second question (labeled 10): "Given the graph below, which of the following represents the shaded region in the graph?"

Step1: Identify function and integration limits from graph.

Function $f(t)=2t^3-2t+3$. Limits from graph: $a=-1, b=0.5$. $f(t) = 2t^3 - 2t + 3$, $a = -1$, $b = 0.5$

Step2: Write the definite integral for the area.

Area is $\int_a^b f(t) dt$. Substitute $f(t)$, $a$, $b$. $$ \text{Area} = \int_{-1}^{0.5} (2t^3 - 2t + 3) dt $$

Step3: Select the matching option.

Compare derived integral with choices. Option 3 matches. Option 3: $\int_{-1}^{0.5} (2t^3 - 2t + 3) dt$

Answer:

Only B) $\int_{-1}^{0.5} (2t^3 - 2t + 3) dt$