Calculus 3: Finding Current Rate of Change in Electric Circuits Using Chain Rule

In summary, Ohm's law states that V = IR in a simple electric circuit, where V is the voltage, I is the current, and R is the resistance. The voltage decreases at a rate of 0.03 volts per second as the battery wears out, and the resistance increases at a rate of 0.02 ohms per second as the resistor heats up. To find the rate of change of the current, we need to take the derivative of I with respect to time. This is represented by dI/dt.
  • #1
krtica
50
0
In a simple electric circuit, Ohm's law states that V = IR, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.03 volts per second and, as the resistor heats up, the resistance is increasing at 0.02 ohms per second. When the resistance is 400 ohms and the current is 0.04 amperes, at what rate is the current changing?

Would I differentiate implicitly with respect to the current, I?
 
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  • #2
start out by differentiating both sides with respect to time, try using the product rule on the right hand side.
 
  • #3
To expand on what tt2348 said, you would NOT differentiate with respect to I. The question is "at what rate is the current changing?" This implies that you should take the derivative of I with respect to something (an excellent candidate would be t), not differentiate with respect to I.

IOW, they're asking for dI/dt, the time rate of change of current.
 

Related to Calculus 3: Finding Current Rate of Change in Electric Circuits Using Chain Rule

1. What is the Chain Rule in Calculus 3?

The Chain Rule is a method used to find the derivative of a composite function, which is a function that is made up of multiple functions inside of it. It allows us to find the rate of change of one variable with respect to another variable in a complex function.

2. How do you use the Chain Rule to find derivatives?

To use the Chain Rule, you first need to identify the inner and outer functions in the composite function. Then, you take the derivative of the outer function and multiply it by the derivative of the inner function. This will give you the derivative of the composite function.

3. Can you provide an example of using the Chain Rule?

Sure, let's say we have the function f(x) = (x^2 + 3)^2. The inner function is x^2 + 3 and the outer function is x^2. To find the derivative, we use the Chain Rule and take the derivative of the outer function (2x) and multiply it by the derivative of the inner function (2x). This gives us the derivative of f(x) = 4x(x^2 + 3).

4. What is the importance of the Chain Rule in Calculus 3?

The Chain Rule is important because it allows us to find the derivative of complex functions, which are common in many real-world applications. It also helps us understand the relationship between different variables in a function and how they change with respect to each other.

5. Are there any common mistakes students make when using the Chain Rule?

One common mistake is forgetting to take the derivative of the inner function. Another mistake is using the derivative of the outer function instead of the inner function. It's important to pay attention to the order in which the functions are composed to avoid these mistakes.

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