Consider the function
ƒ(t) = { t      if  ―2 ≤ t < 0
t²/2    if    0 ≤ t ≤ 2                                                                                                                                                                       
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(f) Find a constant C such that F(𝓍) = G(𝓍) + C .

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding a constant C such that F(𝓍) = G(𝓍) + C, where F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt. This involves comparing the two integrals and determining the relationship between them.
2. Step 2: Analyze the difference between the limits of integration for F(𝓍) and G(𝓍). Note that the lower limit of integration for F(𝓍) is -1, while for G(𝓍) it is -2. This means that G(𝓍) includes an additional integral from t = -2 to t = -1 compared to F(𝓍).
3. Step 3: Compute the integral of ƒ(t) from t = -2 to t = -1. From the graph and the piecewise definition of ƒ(t), ƒ(t) = t for -2 ≤ t < 0. Therefore, the integral from -2 to -1 is ∫₋₂⁻¹ t dt. Use the formula for the integral of t: ∫ t dt = t²/2.
4. Step 4: Evaluate the definite integral ∫₋₂⁻¹ t dt. Substitute the limits of integration into the formula t²/2: [(-1)²/2 - (-2)²/2]. This calculation gives the constant value that represents the difference between F(𝓍) and G(𝓍).
5. Step 5: Conclude that the constant C is equal to the value of the integral from t = -2 to t = -1. Thus, F(𝓍) = G(𝓍) + C, where C is the result of the definite integral computed in Step 4.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function ƒ(t) has two distinct definitions: one for t in the interval [−2, 0) and another for t in the interval [0, 2]. Understanding how to evaluate piecewise functions is crucial for analyzing their behavior across different intervals.

Definite Integrals

Definite integrals calculate the area under a curve between two specified limits. In the question, F(x) and G(x) are defined as integrals of the function ƒ(t) over different intervals. Grasping the concept of definite integrals is essential for determining the relationship between F(x) and G(x) and finding the constant C.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is F(b) - F(a). This theorem is vital for solving the problem, as it allows us to relate the integrals F(x) and G(x) to their respective areas under the curve of ƒ(t) and find the constant C.