Antiderivatives (where did I go wrong?)

• 01010011
In summary, the correct answer for f(t) is t^2 - 3 cos t + 2. The mistake was made in the calculation of the constant C, which should have been -3 instead of +3. The correct value for C is 2.
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Where did I go wrong with my working? The answer in the book is
f(t) = t^2 + 3 cos t + 2

1. The problem statement: Find f for
f prime (t) = 2t - 3 sin t, f(0) = 5

2. Homework Equations :
Most General antiderivation: F(x) + C
Antidifferentiation formula: Function = sin x, Particular antiderivation = -cos x

3. My attempt at a solution

= [(2t^2)/2] -3 (-cos) t + C

= t^2 + 3 cos t + C

f(0) = t^2 + 3 cos t + C = 5

Therefore C = 5 - (0)^2 + 3 cos (0)
C = 5

Therefore f(t) = t^2 - 3 cos t + 5

Cos of 0 equals 1 so 3 cos0=3

And you subtracted 3*cos0 so it needs to be negative not positive.

Dustinsfl said:
Cos of 0 equals 1 so 3 cos0=3

Thank you so much, I understand now!

Dustinsfl said:
And you subtracted 3*cos0 so it needs to be negative not positive.

C = 5 - (0)^2 + 3 cos (0)
C = 5 - 0 - 3
C = 2

Ok, great , thanks

Good except you still have +3 cos but the next steps are you have the negative sign

Dustinsfl said:
Good except you still have +3 cos but the next steps are you have the negative sign

Careless mistakes on my part.

C = 5 - (0)^2 - 3 cos (0)
C = 5 -(3 * 1)
C = 5 - 3
C = 2

1. What is an antiderivative?

An antiderivative is a mathematical function that represents the reverse of a derivative. It is also known as the indefinite integral and is used to find the original function from its derivative.

2. How do I find the antiderivative of a function?

To find the antiderivative of a function, you can use the reverse of the power rule. For example, if the derivative of a function is 3x^2, the antiderivative would be x^3 + C, where C is a constant. There are also other techniques, such as integration by parts and substitution, that can be used to find antiderivatives.

3. Can I use antiderivatives to solve real-world problems?

Yes, antiderivatives have many practical applications in physics, engineering, economics, and other fields. They can be used to find the area under a curve, determine displacement or velocity, and solve optimization problems, to name a few.

4. Why is there a constant (C) in the antiderivative?

The constant (C) in the antiderivative represents all possible solutions to the original function, as the derivative of a constant is always 0. It is important to include the constant when finding the antiderivative, as it allows for an infinite number of possible solutions.

5. What are some common mistakes when finding antiderivatives?

Some common mistakes include forgetting to include the constant (C) in the antiderivative, applying the power rule incorrectly, and not using the correct integration techniques. It is also important to be aware of any specific rules or exceptions for certain functions, such as trigonometric functions or logarithmic functions.

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