If [tex]p(t)=\int_0^{t^4} \sqrt{14+w^6} dw[/tex] then [tex]p^{\prime}(t)=[/tex] ?
Answer (2)
To find the derivative of p ( t ) = ∫ 0 t 4 14 + w 6 d w , we use the Fundamental Theorem of Calculus and the chain rule. The final result is p ′ ( t ) = 4 t 3 14 + t 24 .
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Let p ( t ) = ∫ 0 t 4 14 + w 6 d w .
Apply the substitution u = t 4 .
Use the Fundamental Theorem of Calculus to find d u d p = 14 + u 6 .
Apply the chain rule: p ′ ( t ) = d u d p ⋅ d t d u = 14 + u 6 ⋅ 4 t 3 .
Substitute back u = t 4 to get p ′ ( t ) = 4 t 3 14 + t 24 .
The final answer is 4 t 3 14 + t 24 .
Explanation
Problem Analysis We are given the function p ( t ) = ∫ 0 t 4 14 + w 6 d w and we want to find its derivative p ′ ( t ) . This problem requires us to use the Fundamental Theorem of Calculus and the chain rule.
Substitution Let u = t 4 . Then we can rewrite p ( t ) as p ( t ) = ∫ 0 u 14 + w 6 d w .
Applying the Fundamental Theorem of Calculus By the Fundamental Theorem of Calculus, we have d u d p = 14 + u 6 .
Finding du/dt Now, we need to find d t d u . Since u = t 4 , we have d t d u = 4 t 3 .
Applying the Chain Rule Using the chain rule, we have p ′ ( t ) = d u d p ⋅ d t d u . Substituting the expressions we found, we get p ′ ( t ) = 14 + u 6 ⋅ 4 t 3 .
Final Substitution Finally, substitute u = t 4 back into the expression: p ′ ( t ) = 14 + ( t 4 ) 6 ⋅ 4 t 3 = 4 t 3 14 + t 24 .
Final Answer Therefore, the derivative of p ( t ) is p ′ ( t ) = 4 t 3 14 + t 24 .
Examples
Understanding the Fundamental Theorem of Calculus is essential in many areas of physics and engineering. For example, it can be used to determine the rate of change of heat in a system, the velocity of an object given its acceleration, or the flow rate of a fluid. In essence, whenever you need to find how a cumulative quantity changes, the Fundamental Theorem of Calculus provides the tools to do so.
