Calculus problem differentiation.

In summary, the conversation discusses a calculus problem and the poster's confusion with the solution found online. They ask for clarification on the values and their relationship, as well as how to approach the problem. The moderator offers guidance on using differentiation and clarifies the difference between V(t) and V'(t).
  • #1
Lejas90210
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<Moderator's note: Member has been warned to show some effort before an answer can be given.>

Hello all.
This is my first post in this forum, I am asking for your understanding. I have a problem with the calculus task and I stuck in a dead endso I managed to find a solution on the internet. I am not sure whether it is done correctly. My main questions are:
1.) Why 9.5 * e ^ -1 = 3.495?
2.) Are the values in the table correct? What is the dependence between those values.
3.) If in response to t = 10, -> V (t) = 3.495 and in the table for t = 10 there is value: 60.05 V?
 

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  • #2
:welcome:
## e^{-1}=\frac{1}{2.7828} \approx.36788 ##.
## V'(10)=3.495 ##. (##Note: V'(t)=\frac{dV(t)}{dt} ##). That's the slope of the curve of ## V(t) ## vs. ## t ## at ## t=10 ## if you draw a tangent line. Note: The graph needs to have the y-increment the same as the x-increment to readily see this. (With your increments of 10 and 5, it will appear to have a slope of 3.495/2 at ## t=10 ##).
 
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  • #3
My Problem (Task) require use different rule of differentiation that's what I have till now:
 

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  • #4
You need to realize that ## \frac{dV(t)}{dt} ## is called ## V'(t) ## and is completely different from ## V(t) ##.
If ## V(t) ## were distance, ## V'(t) ## would be the velocity.
## V(t) \neq V'(t) ##. They are two separate functions.
Again ## V'(t)=\frac{d V(t)}{dt} ##.
You can let ## V=y ## , and ## t=x ##, but you should specify this if you chose to take ##V'=y'= \frac{dy}{dx} ##. You then substitute ## V ## (or ## V' ##) and ## t ## back in, to get ## V'(t) ##.
It is good to be systematic rather than have hand-waving in your steps.
 
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What is differentiation in calculus?

Differentiation is a mathematical process used to find the rate of change of a function with respect to its independent variable. It involves finding the derivative of a function, which represents the slope of the function at a specific point.

What are the basic rules of differentiation?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. These rules provide a systematic way to find the derivative of a function by applying specific formulas.

Why is differentiation important in calculus?

Differentiation is important in calculus because it allows us to analyze the behavior of functions and solve various real-world problems. It is also a fundamental concept in calculus, as it is used to find the maximum and minimum values of a function and to determine the shape of a graph.

What is the difference between differentiation and integration?

Differentiation and integration are inverse processes in calculus. Differentiation is used to find the rate of change of a function, while integration is used to find the area under a curve. In other words, differentiation is a process of finding the slope of a function, while integration is a process of finding the area under a curve.

How can differentiation be applied in real life?

Differentiation has many real-life applications, such as in physics, economics, and engineering. For example, it can be used to find the velocity and acceleration of an object, the marginal cost and revenue in economics, and the optimal design of structures in engineering.

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