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

Su sıcaklığının dere alabalığının büyüme hızını etkilediğini göz önünde bulundurun. Belirli bir süre içinde çeşitli su sıcaklıklarında kazandığı kilo miktarı tablolanmıştır. Büyüme hızının sıcaklık değişimlerine yanıtı türevler kullanılarak tahmin edilir.İlk veri noktasında, türev, birinci ve ikinci noktalar arasında çizilen ileri bir sekant çizgisinin eğimi kullanılarak yaklaşık olarak hesaplanır.İkinci noktada, türev, önceki ve sonraki noktaları birleştiren sekant çizgisinin eğimi hesaplanarak merkezi bir fark kullanılarak tahmin edilir; bu da türevin daha doğru bir tahminini sağlar.Üçüncü noktada, yine merkezi bir fark kullanılarak önceki ve sonrakı noktaları birbirine bağlayan sekant çizgisinin eğimini hesaplayın.Benzer şekilde, dördüncü noktada, türev, komşu veri noktaları arasındaki sekant çizgisinin eğimi kullanılarak hesaplanır.Son olarak, son noktada, türev, son iki veri noktası arasında çekilen geriye doğru kesik çizgisinin eğimi kullanılarak yaklaşık olarak hesaplanır.Bu türev tahminler bir tabloda derlenir. İlgili grafik, büyüme hızında tutarlı bir düşüş göstermekte olup, sıcaklık ile kilo artışı arasında negatif bir korelasyon göstermektedir.

değişim oranı: problem Çözme

Rate of Change: Problem Solving

Dere Alabal&#x131;&#x11F;&#x131;n&#x131;n S&#x131;cakl&#x131;&#x11F;a Ba&#x11F;l&#x131; B&#xFC;y&#xFC;mesi
Dere alabal&#x131;&#x11F;&#x131;n&#x131;n b&#xFC;y&#xFC;mesi su s&#x131;cakl&#x131;&#x11F;&#x131;ndan do&#x11F;rudan etkilenir. Deneysel veriler, alabal&#x131;k a&#x11F;&#x131;rl&#x131;&#x11F;&#x131;n&#x131;n de&#x11F;i&#x15F;en su s&#x131;cakl&#x131;klar&#x131;na ba&#x11F;l&#x131; olarak 24 g&#xFC;nl&#xFC;k s&#xFC;re boyunca nas&#x131;l de&#x11F;i&#x15F;ti&#x11F;ini g&#xF6;stermektedir. 15,5 &#xB0;C gibi daha d&#xFC;&#x15F;&#xFC;k s&#x131;cakl&#x131;klarda dere alabal&#x131;&#x11F;&#x131; &#xF6;nemli bir a&#x11F;&#x131;rl&#x131;k art&#x131;&#x15F;&#x131; sergiler. Ancak s&#x131;cakl&#x131;k artt&#x131;k&#xE7;a a&#x11F;&#x131;rl&#x131;k kazan&#x131;m&#x131; giderek azal&#x131;r. &#xD6;l&#xE7;&#xFC;len en y&#xFC;ksek s&#x131;cakl&#x131;k olan 24,4 &#xB0;C&#x2019;de alabal&#x131;klar net a&#x11F;&#x131;rl&#x131;k kayb&#x131; ya&#x15F;ar. Bu &#xF6;r&#xFC;nt&#xFC;, dere alabal&#x131;&#x11F;&#x131;n&#x131;n daha serin sularda daha iyi geli&#x15F;ti&#x11F;ini ve daha s&#x131;cak ko&#x15F;ullarda metabolik stres ya da b&#xFC;y&#xFC;me verimlili&#x11F;inin azalmas&#x131;yla kar&#x15F;&#x131;la&#x15F;t&#x131;&#x11F;&#x131;n&#x131; d&#xFC;&#x15F;&#xFC;nd&#xFC;rmektedir.
B&#xFC;y&#xFC;me Oran&#x131;ndaki De&#x11F;i&#x15F;imlerin Tahmini
A&#x11F;&#x131;rl&#x131;k art&#x131;&#x15F; oran&#x131;n&#x131;n s&#x131;cakl&#x131;kla nas&#x131;l de&#x11F;i&#x15F;ti&#x11F;ini anlamak i&#xE7;in bilim insanlar&#x131;, ard&#x131;&#x15F;&#x131;k s&#x131;cakl&#x131;k de&#x11F;erleri aras&#x131;ndaki a&#x11F;&#x131;rl&#x131;k de&#x11F;i&#x15F;im oran&#x131;n&#x131; tahmin eder. Bu i&#x15F;lem, k&#xFC;&#xE7;&#xFC;k s&#x131;cakl&#x131;k aral&#x131;klar&#x131; i&#xE7;inde a&#x11F;&#x131;rl&#x131;ktaki de&#x11F;i&#x15F;imi kar&#x15F;&#x131;la&#x15F;t&#x131;rarak yap&#x131;l&#x131;r. Yaln&#x131;zca tek bir kom&#x15F;u s&#x131;cakl&#x131;k noktas&#x131; bulundu&#x11F;unda tahmin bu iki nokta aras&#x131;ndaki de&#x11F;i&#x15F;ime dayan&#x131;r. Bir veri noktas&#x131; iki di&#x11F;er veri noktas&#x131; aras&#x131;nda yer ald&#x131;&#x11F;&#x131;nda ise her iki taraftaki de&#x11F;i&#x15F;im oranlar&#x131;n&#x131;n ortalamas&#x131;n&#x131; hesaplamak daha iyi bir tahmin sa&#x11F;lar. Bu de&#x11F;erler, alabal&#x131;&#x11F;&#x131;n b&#xFC;y&#xFC;me tepkisinin s&#x131;cakl&#x131;kla ne kadar h&#x131;zl&#x131; de&#x11F;i&#x15F;ti&#x11F;i konusunda &#xE7;&#x131;kar&#x131;m yapmay&#x131; sa&#x11F;lar.
B&#xFC;y&#xFC;me H&#x131;z&#x131; E&#x11F;ilimlerinin Yorumlanmas&#x131;
Elde edilen tahminler, s&#x131;cakl&#x131;k artt&#x131;k&#xE7;a a&#x11F;&#x131;rl&#x131;k art&#x131;&#x15F; oran&#x131;n&#x131;n daha negatif de&#x11F;erlere kayd&#x131;&#x11F;&#x131;n&#x131; ortaya koymaktad&#x131;r. Bu durum, dere alabal&#x131;klar&#x131;n&#x131;n yaln&#x131;zca daha d&#xFC;&#x15F;&#xFC;k verimle b&#xFC;y&#xFC;d&#xFC;&#x11F;&#xFC;n&#xFC; de&#x11F;il, daha y&#xFC;ksek s&#x131;cakl&#x131;klarda nihayet a&#x11F;&#x131;rl&#x131;k kaybetmeye ba&#x15F;lad&#x131;&#x11F;&#x131;n&#x131; da ortaya koyar. De&#x11F;i&#x15F;im oran&#x131;, s&#x131;cakl&#x131;ktaki her 1 &#xB0;C art&#x131;&#x15F; ba&#x15F;&#x131;na alabal&#x131;k a&#x11F;&#x131;rl&#x131;&#x11F;&#x131;n&#x131;n ne kadar de&#x11F;i&#x15F;ti&#x11F;ini ifade eden g/&#xB0;C birimiyle belirtilir. Bu de&#x11F;erlerin grafi&#x11F;i, istikrarl&#x131; bir azal&#x131;&#x15F; g&#xF6;stererek daha y&#xFC;ksek su s&#x131;cakl&#x131;klar&#x131;n&#x131;n dere alabal&#x131;&#x11F;&#x131;n&#x131;n b&#xFC;y&#xFC;mesi i&#xE7;in giderek daha elveri&#x15F;siz oldu&#x11F;u sonucunu desteklemektedir.

Değişim Oranı: Problem Çözme

Dere Alabal&#x131;&#x11F;&#x131;n&#x131;n S&#x131;cakl&#x131;&#x11F;a Ba&#x11F;l&#x131; B&#xFC;y&#xFC;mesi
Dere alabal&#x131;&#x11F;&#x131;n&#x131;n ...

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JoVE Core

Calculus

Kalkülüs

Unit

Birim

Derivatives

Türevler

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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Rates of change describe how one variable changes in response to another. In math, this idea helps model dynamic systems in physics, biology, economics, and engineering. It also helps students study real-world functions that change over time or across different inputs. The average rate of change measures the overall change in a function over an interval [x1, x2]. It is found by dividing the change in output by the change in input. On a graph, this value is the slope of the secant line, which...

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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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