Math  /  Calculus

QuestionConsider the function f(x)=10x10+6x73x36f(x)=10 x^{10}+6 x^{7}-3 x^{3}-6. An antiderivative of f(x)f(x) is F(x)=Axn+Bxm+Cxp+DxqF(x)=A x^{n}+B x^{m}+C x^{p}+D x^{q} where AA is \square and nn is \square and BB is \square and mm is \square and CC is \square and pp is \square and DD is \square and qq is \square

Studdy Solution

STEP 1

What is this asking? We're looking for a function F(x)F(x) whose derivative is the given function f(x)f(x).
It's like playing detective, but with math! Watch out! Don't forget about the power rule and how constants behave when you take derivatives!
Also, remember that an antiderivative isn't unique; you can always add a constant!

STEP 2

1. Find the antiderivative of each term.
2. Combine the antiderivatives.

STEP 3

Let's start with the first term of f(x)f(x), which is 10x1010x^{10}.
We're looking for a term whose derivative is 10x1010x^{10}.
Remember the power rule?
When you take the derivative, you multiply by the exponent and then decrease the exponent by 1.
So, to go backward (find the antiderivative), we do the opposite!
We'll increase the exponent by 1 and divide by the new exponent.

STEP 4

So, for 10x1010x^{10}, we increase the exponent to **11** and divide by **11**.
This gives us 1011x11\frac{10}{11}x^{11}.
Let's check: the derivative of 1011x11\frac{10}{11}x^{11} is 101111x10=10x10\frac{10}{11} \cdot 11x^{10} = 10x^{10}.
Perfect!

STEP 5

Now, let's do the same for 6x76x^7.
Increase the exponent to **8** and divide by **8**, giving us 68x8=34x8\frac{6}{8}x^8 = \frac{3}{4}x^8.
Check: the derivative of 34x8\frac{3}{4}x^8 is 348x7=6x7\frac{3}{4} \cdot 8x^7 = 6x^7.
Awesome!

STEP 6

Next up: 3x3-3x^3.
Increase the exponent to **4** and divide by **4**, giving us 34x4-\frac{3}{4}x^4.
Check: the derivative of 34x4-\frac{3}{4}x^4 is 344x3=3x3-\frac{3}{4} \cdot 4x^3 = -3x^3.
Looking good!

STEP 7

Finally, we have 6-6.
Think of this as 6x0-6x^0.
Increasing the exponent by 1 gives us 6x1-6x^1, or just 6x-6x.
Check: the derivative of 6x-6x is 6-6.
Fantastic!

STEP 8

Now, we put it all together!
Adding up the antiderivatives of each term gives us the antiderivative of the whole function: F(x)=1011x11+34x834x46xF(x) = \frac{10}{11}x^{11} + \frac{3}{4}x^8 - \frac{3}{4}x^4 - 6x.

STEP 9

So, we have A=1011A = \frac{10}{11} and n=11n = 11, B=34B = \frac{3}{4} and m=8m = 8, C=34C = -\frac{3}{4} and p=4p = 4, and D=6D = -6 and q=1q = 1.
Remember, we could also add any constant to F(x)F(x) and it would still be an antiderivative!

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