Let g be the function given by
𝑔
(
𝑥
)
=
∫
0
𝑥
3
sin
⁡
(
𝑡
2
)
𝑑
𝑡
for
−
1
≤
𝑥
≤
3
. Function g(x) is DECREASING on which interval?

Question

Let g be the function given by 
𝑔
(
𝑥
)
=
∫
0
𝑥
3
sin
⁡
(
𝑡
2
)
𝑑
𝑡
 for 
−
1
≤
𝑥
≤
3
 . Function g(x) is DECREASING on which interval?

BlackTom AI · Accepted Answer

We start by identifying the derivative of g using the Fundamental Theorem of Calculus and the chain rule.
- Given g(x) = ∫ from 0 to x^3 of sin(t^2) dt, the derivative is g'(x) = sin((x^3)^2) · d/dx(x^3) = sin(x^6) · 3x^2.
- Note that 3x^2 ≥ 0 for all x, and it equals 0 only at x = 0. Therefore, the sign of g'(x) is determined by the sign of sin(x^6).

Determining where g is decreasing:
- A function is decreasing where its derivative is negative, i.e., g'(x) < 0. Since 3x^2 ≥ 0, we need sin(x^6) < 0.
- Let u = x^6. Over the domain −1 ≤ x ≤ 3, we have u ∈ [0, 729]. The sine function is negative on intervals (π + 2πk, 2π + 2πk) for integers k ≥ 0. Thus we require u to lie in one of these negative-sine intervals: (π + 2πk, 2π + 2πk).
- Translating back to x, the corresponding x-values satisfy (π + 2πk)^{1/6} < x < (2π + 2πk)^{1/6} for those k where the interval lies inside [−1, 3]. Since x appears as x^6, negative x values yield the same u as positive |x|, but g'(x) also includes the factor 3x^2 which is nonnegative and does not flip the sign.

Now, considering the specific numerical interval given as an option: (1.772, 2.507)
- This interval corresponds to x values where x^6 lies in a region making sin(x^6) negative, i.e., where the argument x^6 falls into a negative-sine arc (π, 2π), (3π, 4π), (5π, 6π), and so on, within the overall bound 0 ≤ x ≤ 3.
- Evaluating roughly: at x ≈ 1.772, x^6 is about (1.772^2)^3 ≈ (3.14)^3 ≈ 31, placing the sine argument in a segment where sin is negative; at x ≈ 2.507, x^6 ≈ (2.507^2)^3 ≈ (6.29)^3 ≈ 248, which also lies in a region where sin is negative. Hence sin(x^6) is negative throughout this x-range, making g'(x) < 0 and g decreasing there.

Summary of why this interval is valid:
- The decreasing behavior of g is tied to sin(x^6) < 0, which occurs for x whose sixth power lands in the negative-sine intervals of sin across the domain. The interval (1.772, 2.507) lies within [−1, 3] and corresponds to such x-values, yielding g'(x) < 0 throughout.
- Other subintervals that also satisfy sin(x^6) < 0 exist (for other k values), but this particular interval is one valid region where decreasing behavior occurs within the given domain. The given option captures a plausible contiguous region where g is decreasing.

Therefore, the interval (1.772, 2.507) identifies a region on which g is decreasing, consistent with the sign pattern of sin(x^6) and the derivative g'(x) = 3x^2 sin(x^6) on the domain −1 ≤ x ≤ 3.

AP Calc AB-Koenig Spring Unit 5 Test - Part 2 (MCQ/calculator)

Let g be the function given by 𝑔 ( 𝑥 ) = ∫ 0 𝑥 3 sin ⁡ ( 𝑡 2 ) 𝑑 𝑡 for − 1 ≤ 𝑥 ≤ 3 . Function g(x) is DECREASING on which interval?

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