Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

c. Find a formula for A(x)

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

검증된 단계별 안내

Step 1: Understand the problem. We need to find a formula for A(x), which represents the area under the curve y = f(t) from t = 0 to t = x. The function f(t) is piecewise, meaning it has different expressions depending on the value of t.

Step 2: Identify the intervals. The function f(t) is defined as -2t + 8 for t ≤ 3 and 2 for t > 3. Therefore, we need to consider two cases: when 0 ≤ x ≤ 3 and when x > 3.

Step 3: Calculate A(x) for 0 ≤ x ≤ 3. In this interval, f(t) = -2t + 8. The area A(x) is the integral of f(t) from 0 to x. Set up the integral: A(x) = ∫ from 0 to x of (-2t + 8) dt.

Step 4: Calculate A(x) for x > 3. In this interval, we need to consider the area from 0 to 3 using the first part of the function and from 3 to x using the second part. First, find the area from 0 to 3: A(3) = ∫ from 0 to 3 of (-2t + 8) dt. Then, add the area from 3 to x: ∫ from 3 to x of 2 dt.

Step 5: Combine the results. For 0 ≤ x ≤ 3, A(x) is the result of the integral from Step 3. For x > 3, A(x) is the sum of the area from 0 to 3 and the integral from 3 to x from Step 4. This gives us a piecewise formula for A(x).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integral

The definite integral of a function over an interval gives the net area between the function's graph and the x-axis. In this context, to find the area function A(x), we need to compute the definite integral of f(t) from 0 to x. This process involves evaluating the integral based on the piecewise definition of the function f(t) provided.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(t) has two distinct expressions: one for t ≤ 3 and another for t > 3. Understanding how to handle piecewise functions is crucial for correctly calculating the area A(x) since the formula will change depending on whether x is less than, equal to, or greater than 3.

Area Under a Curve

The area under a curve represents the accumulation of values of a function over a specified interval. For the function f(t), the area A(x) can be interpreted as the total area from the t-axis to the curve from t=0 to t=x. This concept is fundamental in calculus, as it connects geometric interpretations with integral calculus, allowing us to quantify the area based on the function's behavior.

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.c. Find a formula for A(x)ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>