Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.R.35

Capitolo 5, Problema 5.R.35

Find the intervals on which ƒ(𝓍) = ∫ₓ¹ (t―3) (t―6)¹¹ dt is increasing and the intervals on which it is decreasing.

Guida verificata passo dopo passo

Step 1: Recognize that the function ƒ(𝓍) is defined as a definite integral with a variable upper limit. To determine where ƒ(𝓍) is increasing or decreasing, we need to compute its derivative using the Fundamental Theorem of Calculus.

Step 2: Apply the Fundamental Theorem of Calculus, which states that if ƒ(𝓍) = ∫ₓ¹ g(t) dt, then ƒ'(𝓍) = -g(𝓍). Here, g(t) = (t - 3)(t - 6)¹¹, so ƒ'(𝓍) = -(𝓍 - 3)(𝓍 - 6)¹¹.

Step 3: Analyze the sign of ƒ'(𝓍) to determine where ƒ(𝓍) is increasing or decreasing. ƒ(𝓍) is increasing when ƒ'(𝓍) > 0 and decreasing when ƒ'(𝓍) < 0. This requires solving the inequality -(𝓍 - 3)(𝓍 - 6)¹¹ > 0 and -(𝓍 - 3)(𝓍 - 6)¹¹ < 0.

Step 4: Examine the critical points of ƒ'(𝓍). The factors (𝓍 - 3) and (𝓍 - 6)¹¹ determine the behavior of ƒ'(𝓍). Note that (𝓍 - 6)¹¹ is always non-negative because it is raised to an odd power. The sign of ƒ'(𝓍) depends on the factor -(𝓍 - 3).

Step 5: Determine the intervals of increase and decrease by testing the sign of ƒ'(𝓍) in the intervals divided by the critical points 𝓍 = 3 and 𝓍 = 6. Summarize the intervals where ƒ'(𝓍) > 0 (increasing) and ƒ'(𝓍) < 0 (decreasing).

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval gives the net change of the function. This theorem allows us to evaluate the integral and find the function's behavior based on its derivative.

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Fundamental Theorem of Calculus Part 1

Derivative and Increasing/Decreasing Functions

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if its derivative is negative. By analyzing the sign of the derivative, we can determine where the function is rising or falling, which is essential for solving the given problem regarding the intervals of increase and decrease.

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Determining Where a Function is Increasing & Decreasing

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are crucial for determining the intervals of increase and decrease, as they can indicate potential local maxima or minima. By evaluating the derivative at these points, we can ascertain the behavior of the function around them.

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Critical Points

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