Math / Calculus

QuestionFor the following function $f$ , find the antiderivative $F$ that satisfies the given condition. $f(x)=6 \sqrt{x}+4 ; F(1)=4$
The antiderivative that satisfies the given condition is $F(x)=4 x \sqrt{x}+4 x+c$ .

Studdy Solution

STEP 1

What is this asking? We need to find a function $F(x)$ whose derivative is $f(x) = 6\sqrt{x} + 4$ and also makes sure $F(1) = 4$ . Watch out! Don't forget about the "+ C" when finding antiderivatives, and make sure to use the given condition to find the specific value of C!

STEP 2

1. Rewrite the function
2. Find the general antiderivative
3. Solve for C

STEP 3

Let's rewrite the function $f(x)$ using exponents instead of the square root symbol to make it easier to work with.
Remember that $\sqrt{x}$ is the same as $x^{\frac{1}{2}}$ .
So, we have $f(x) = 6x^{\frac{1}{2}} + 4$ .
This sets us up perfectly for using the power rule for integration!

STEP 4

Now, let's find the general antiderivative of $f(x)$ .
Remember, the power rule for integration says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n \ne -1$ .

STEP 5

Applying this rule to each term in our rewritten $f(x)$ , we get: $\int (6x^{\frac{1}{2}} + 4) \, dx = 6 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + 4x + C$ Simplifying the exponents and denominators gives us: $6 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + 4x + C$ Dividing by a fraction is the same as multiplying by its reciprocal, so we have: $6 \cdot \frac{2}{3} x^{\frac{3}{2}} + 4x + C$ Multiplying the constants gives us the general antiderivative: $F(x) = 4x^{\frac{3}{2}} + 4x + C$

STEP 6

We know that $F(1) = 4$ , so let's plug in $x = 1$ into our general antiderivative: $F(1) = 4(1)^{\frac{3}{2}} + 4(1) + C = 4$ Since $1$ raised to any power is still $1$ , we get: $4 + 4 + C = 4$ $8 + C = 4$

STEP 7

To isolate $C$ , we subtract $8$ from both sides of the equation: $C = 4 - 8$ $C = -4$

STEP 8

So, our specific antiderivative is $F(x) = 4x^{\frac{3}{2}} + 4x - 4$ .
Remember, $x^{\frac{3}{2}}$ is the same as $x\sqrt{x}$ , so we can also write it as $F(x) = 4x\sqrt{x} + 4x - 4$ .

STEP 9

The antiderivative that satisfies the given condition is $F(x) = 4x\sqrt{x} + 4x - 4$ .

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QuestionFor the following function $f$ , find the antiderivative $F$ that satisfies the given condition. $f(x)=6 \sqrt{x}+4 ; F(1)=4$
The antiderivative that satisfies the given condition is $F(x)=4 x \sqrt{x}+4 x+c$ .

Studdy Solution

STEP 1

What is this asking? We need to find a function $F(x)$ whose derivative is $f(x) = 6\sqrt{x} + 4$ and also makes sure $F(1) = 4$ . Watch out! Don't forget about the "+ C" when finding antiderivatives, and make sure to use the given condition to find the specific value of C!

STEP 2

1. Rewrite the function
2. Find the general antiderivative
3. Solve for C

STEP 3

Let's rewrite the function $f(x)$ using exponents instead of the square root symbol to make it easier to work with.
Remember that $\sqrt{x}$ is the same as $x^{\frac{1}{2}}$ .
So, we have $f(x) = 6x^{\frac{1}{2}} + 4$ .
This sets us up perfectly for using the power rule for integration!

STEP 4

Now, let's find the general antiderivative of $f(x)$ .
Remember, the power rule for integration says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n \ne -1$ .

STEP 5

STEP 6

We know that $F(1) = 4$ , so let's plug in $x = 1$ into our general antiderivative: $F(1) = 4(1)^{\frac{3}{2}} + 4(1) + C = 4$ Since $1$ raised to any power is still $1$ , we get: $4 + 4 + C = 4$ $8 + C = 4$

STEP 7

To isolate $C$ , we subtract $8$ from both sides of the equation: $C = 4 - 8$ $C = -4$

STEP 8

So, our specific antiderivative is $F(x) = 4x^{\frac{3}{2}} + 4x - 4$ .
Remember, $x^{\frac{3}{2}}$ is the same as $x\sqrt{x}$ , so we can also write it as $F(x) = 4x\sqrt{x} + 4x - 4$ .

STEP 9

The antiderivative that satisfies the given condition is $F(x) = 4x\sqrt{x} + 4x - 4$ .

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QuestionFor the following function fff, find the antiderivative FFF that satisfies the given condition. f(x)=6x+4;F(1)=4f(x)=6 \sqrt{x}+4 ; F(1)=4f(x)=6x​+4;F(1)=4The antiderivative that satisfies the given condition is F(x)=4xx+4x+cF(x)=4 x \sqrt{x}+4 x+cF(x)=4xx​+4x+c.

Studdy Solution

STEP 1

STEP 2

1. Rewrite the function2. Find the general antiderivative3. Solve for C

STEP 3

STEP 4

Now, let's **find** the general antiderivative of f(x)f(x)f(x).Remember, the power rule for integration says: ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C, where n≠−1n \ne -1n=−1.

STEP 5

STEP 6

We know that F(1)=4F(1) = 4F(1)=4, so let's **plug in** x=1x = 1x=1 into our general antiderivative: F(1)=4(1)32+4(1)+C=4 F(1) = 4(1)^{\frac{3}{2}} + 4(1) + C = 4 F(1)=4(1)23​+4(1)+C=4 Since 111 raised to any power is still 111, we get: 4+4+C=4 4 + 4 + C = 4 4+4+C=4 8+C=4 8 + C = 4 8+C=4

STEP 7

To **isolate** CCC, we subtract 888 from both sides of the equation: C=4−8 C = 4 - 8 C=4−8 C=−4 C = -4 C=−4

STEP 8

So, our **specific antiderivative** is F(x)=4x32+4x−4F(x) = 4x^{\frac{3}{2}} + 4x - 4F(x)=4x23​+4x−4.Remember, x32x^{\frac{3}{2}}x23​ is the same as xxx\sqrt{x}xx​, so we can also write it as F(x)=4xx+4x−4F(x) = 4x\sqrt{x} + 4x - 4F(x)=4xx​+4x−4.

STEP 9

The antiderivative that satisfies the given condition is F(x)=4xx+4x−4F(x) = 4x\sqrt{x} + 4x - 4F(x)=4xx​+4x−4.

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QuestionFor the following function $f$ , find the antiderivative $F$ that satisfies the given condition. $f(x)=6 \sqrt{x}+4 ; F(1)=4$
The antiderivative that satisfies the given condition is $F(x)=4 x \sqrt{x}+4 x+c$ .

1. Rewrite the function
2. Find the general antiderivative
3. Solve for C

Now, let's find the general antiderivative of $f(x)$ .
Remember, the power rule for integration says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n \ne -1$ .

We know that $F(1) = 4$ , so let's plug in $x = 1$ into our general antiderivative: $F(1) = 4(1)^{\frac{3}{2}} + 4(1) + C = 4$ Since $1$ raised to any power is still $1$ , we get: $4 + 4 + C = 4$ $8 + C = 4$

To isolate $C$ , we subtract $8$ from both sides of the equation: $C = 4 - 8$ $C = -4$

So, our specific antiderivative is $F(x) = 4x^{\frac{3}{2}} + 4x - 4$ .
Remember, $x^{\frac{3}{2}}$ is the same as $x\sqrt{x}$ , so we can also write it as $F(x) = 4x\sqrt{x} + 4x - 4$ .

The antiderivative that satisfies the given condition is $F(x) = 4x\sqrt{x} + 4x - 4$ .