Name: brook trout growth vs water temperature
Uploaded: 2026-07-24T11:46:38.212Z
Duration: 1 min 26 s
Channel: JoVE

水温がブルックトラウトの成長速度に影響を与えていることを考慮してください。一定期間にわたり、さまざまな水温で増加した重量をまとめています。成長速度の温度変化への応答は微分を用いて推定されます。最初のデータ点では、1点目と2点目の間に引かれた前方割線の傾きを用いて微分を近似します。2点目では、前後の点をつなぐ割線の傾きを計算することで中心差を用いて微分を推定し、より正確な微分推定が得られます。3点目では、再び中心差を使って前後の点を結ぶ割線の傾きを計算します。同様に、4点目では、隣接するデータ点間の割線の傾きを用いて微分を計算します。最後に、最後の点で微分は、最後の2つのデータ点間に描かれた後方割線の傾きを用いて近似します。これらの微分推定は表にまとめられています。対応するプロットは成長率の一貫した低下を示しており、気温と体重増加の間に負の相関が見られます。

変化率：問題解決

Rate of Change: Problem Solving

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Education

JoVE Core

Calculus

微積分学

Unit

単位

Derivatives

導関数

KEY TERMS AND CONCEPTS

主要用語と概念

brook trout growth vs water temperature

Brook trout growth changes with water temperature. The data show how trout weight shifts over a 24-day period at several temperatures. At cooler water temperatures, such as 15.5 degrees Celsius, the trout gain a lot of weight. As the temperature rises, the amount of weight gained gets smaller.

At the warmest temperature tested, 24.4 degrees Celsius, the trout have a net loss in weight. This pattern suggests that brook trout do better in cooler water. Warmer water may create metabolic stress or reduce growth efficiency.

The rate of change in weight is estimated by comparing nearby temperature values. Rate of change means how much the trout's weight changes for each degree Celsius. When only one nearby point is available, the estimate uses the change between those two points. When a point lies between two others, the two nearby rates are averaged to get a better estimate.

These estimates show that the growth rate becomes more negative as temperature increases. That means the trout grow less efficiently and may begin losing weight in warmer water. A graph of the rates shows a steady decline. The overall trend supports the conclusion that higher water temperatures are less favorable for brook trout growth.

Brook Trout Growth vs Water Temperature

tangent line

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derivatives

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Derivatives describe instantaneous rate of change, which is a key idea in both mathematics and physics. They are used to show how a moving object changes position over time. A derivative at a point is also the slope of the tangent line to a curve at that point. It tells how much the output of a function changes for a tiny change in input. For the square root function, finding this rate of change starts with the limit of the difference quotient. The algebra is harder here because the numerator...

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rate of change: problem solving

Brook trout growth changes with water temperature. The data show how trout weight shifts over a 24-day period at several temperatures. At cooler water temperatures, such as 15.5 degrees Celsius, the trout gain a lot of weight. As the temperature rises, the amount of weight gained gets smaller. At the warmest temperature tested, 24.4 degrees Celsius, the trout have a net loss in weight. This pattern suggests that brook trout do better in cooler water. Warmer water may create metabolic stress or...

key terms and concepts

See how brook trout weight changes with water temperature and how rate of change helps show why cooler water supports better growth.

Brook Trout Growth vs Water Temperature | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Brook Trout Growth vs Water Temperature

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