1.8.4 AP Calculus Exam Integral of piece-wise function

In summary, the conversation discusses a problem involving the use of macros in Overleaf, specifically in relation to a mathematical equation. The conversation mentions that observation can solve part (a) of the problem, but parts (b) and (c) are more challenging. The conversation also discusses the use of the fundamental theorem of calculus and the concept of horizontal tangents and inflection points in solving the problem.
  • #1
karush
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image due to macros in Overleaf

ok I think (a) could just be done by observation by just adding up obvious areas

but (b) and (c) are a litte ?

sorry had to post this before the lab closes
 

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  • #2
Is this the intro to the problem you're trying to cite ?

rvfGV0-1024x461.png
 

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  • #3
skeeter said:
Is this the intro to the problem you're trying to cite ?

Yeah I have seen it in several AP books but never did it.
 
  • #4
karush said:
image due to macros in Overleaf

ok I think (a) could just be done by observation by just adding up obvious areas

but (b) and (c) are a litte ?

sorry had to post this before the lab closes

correction ... $\displaystyle g(x) = \int_1^x f(t) \,dt$

(a) $\displaystyle g(2) = \int_1^2 f(t) \,dt = \dfrac{1}{4}$

$\displaystyle g(-2) = \int_1^{-2} f(t) \,dt = -\int_{-2}^1 f(t) \,dt = \dfrac{\pi - 3}{2}$

(b) from the FTC, $g'(x) = f(x) \implies g''(x) = f'(x)$

$g'(3) = f(3) = -1$, $g''(3) = f'(3) = -\dfrac{1}{2}$

(c) horizontal tangents to $g \implies g' = f = 0$

$f = 0$ at $x = \pm 1$ ... relative max at $x=-1$ because $g'$ changes sign from (+) to (-)

neither a max or min at $x=1$ because $g' = f$ does not change sign.

(d) inflection points occur where $g''=f'=0$ or is undefined and $g''=f'$ changes sign.

$g'' = f'$ equals 0 and is undefined at $x = \pm 1$ ... inflection point at $x=1$

$g'' = f'$ is undefined at $x=-2$ ... inflection point there, also because $g''=f'$ changes sign.
 
  • #5
mahalo:cool:

That was a great help...

I did look at some other free responses, but it was messy...:confused:
 

1. What is the format of the 1.8.4 AP Calculus Exam?

The 1.8.4 AP Calculus Exam consists of two sections: multiple-choice and free-response. The multiple-choice section contains 45 questions, while the free-response section has 6 questions. The exam is 3 hours and 15 minutes long.

2. What is a piece-wise function?

A piece-wise function is a mathematical function that is defined by different equations on different intervals. This means that the function has different rules for different parts of its domain.

3. How do you find the integral of a piece-wise function?

To find the integral of a piece-wise function, you need to break it down into its different intervals and use the appropriate integration rules for each interval. Then, you can combine the results to get the overall integral of the function.

4. What is the purpose of finding the integral of a piece-wise function?

The integral of a piece-wise function can be used to find the area under the curve of the function, as well as to solve problems involving rates of change and accumulation. It is an important concept in calculus and has many real-world applications.

5. How can I prepare for the 1.8.4 AP Calculus Exam?

To prepare for the 1.8.4 AP Calculus Exam, it is important to review all of the material covered in the course, including concepts such as derivatives, integrals, and applications of calculus. Practice exams and problems can also help you become familiar with the format of the exam and identify areas where you may need more review.

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