Integrals and Derivatives: Solving Problems with Limits and Rates of Change

In summary, in this conversation, the speaker is seeking help with integrals and derivatives. They are unsure about the correct approach and ask for clarification on certain parts. The conversation includes a discussion of the anti-derivatives for 2sin(x) and 4sec^2(x), as well as the use of the Leibniz-Newton formula and the chain rule of differentiation.
  • #1
SA32
32
0

Homework Statement


In the attachment: 5aef, 6abc


The Attempt at a Solution


5a.) I started by dividing the question into two integrals: the integral of 2sin(x) evaluated from 0 to pi/4 minus the integral of 4sec2(x) evaluated from 0 to pi/4. Then the anti-derivative of 2sin(x) is -2cos(x) and the anti-derivative of 4sec2(x) is 4tan(x). I'm not sure if this is correct, though?

5e&f.) I'm not even sure what this is asking. It looks like "the integral of the derivative", which, according to my notes, is f(x)+C, but... help? I'm not sure.

6abc.) Again, not sure what this is asking. Looks like the derivative of the integral, which is f(x) in my notes. I made an attempt at the first one using the way my friend said to do it, you can see it in the second attachment, but I don't know if it is correct.

Thanks for any help!
 

Attachments

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  • #2
Yes, 5e and 5f are asking for the integral of a derivative, and 6 is asking for the derivative of an integral.
 
  • #3
For 5a) what you're thinking is right so keep doing it.

For 5e) or 5f) use that

[tex] \int_{a}^{b} \frac{df}{dx}{}dx= f(b)-f(a) [/tex]

For 6abc) use the Leibniz-Newton formula and the chain rule of differentiation.

Daniel.
 

What is the difference between an integral and a derivative?

An integral is the reverse operation of a derivative. While a derivative tells us the rate of change of a function at a specific point, an integral tells us the total amount of change over a given interval.

How are integrals and derivatives used in real-world applications?

Integrals and derivatives are used in many fields, such as physics, engineering, economics, and statistics. They are used to model and analyze rates of change, such as velocity and acceleration, and to solve optimization problems.

What is the fundamental theorem of calculus?

The fundamental theorem of calculus states that the integral of a function can be calculated by finding its antiderivative and evaluating it at the upper and lower limits of integration.

What is the relationship between integrals and area under a curve?

Integrals can be used to calculate the area under a curve. The integral of a function represents the sum of infinitely many infinitely thin rectangles, which can be used to approximate the area under the curve.

Can integrals and derivatives be calculated for all functions?

Not all functions have integrals and derivatives. For a function to be differentiable (have a derivative), it must be continuous. Similarly, for a function to be integrable (have an integral), it must be bounded and have a finite number of discontinuities.

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