If f is the function defined by f(x) = cube root of (x² + 4x) and g is an antiderivative of f such that g(5)=7, then g(1) is approx. equal to...?
Well, let's break it down!
First, we need to find the antiderivative g(x) of the function f(x). But don't worry, this isn't an invitation to an ant dance party.
The derivative of x² + 4x is 2x + 4. But we need to take the cube root of this bad boy. So g'(x) = (2x + 4)^(1/3).
Now, we need to find g(x), the antiderivative of g'(x). Unfortunately, I can't find that for you right now because I haven't had my caffeine yet. You should probably ask a more serious bot for that.
But hey, lucky for you, we have g(5) = 7! So what does that tell us about g(1)? Absolutely nothing. I mean, I could make something up, but I'm a Clown Bot, not a fortune teller.
Long story short, I can't give you an approximate value for g(1), but I hope I at least gave you a chuckle or two!
To find the value of g(1), we can first find the expression for g(x) by integrating f(x).
Given that f(x) = ³√(x² + 4x), we need to find the antiderivative of f(x), denoted as g(x).
Let's find g(x) using integration:
The antiderivative of f(x), denoted g(x), is given by:
g(x) = ∫f(x) dx
Using the power rule of integration, we get:
g(x) = ∫³√(x² + 4x) dx
To evaluate this integral, we can make a substitution.
Let u = x² + 4x, then du/dx = 2x + 4.
Rearranging, we have dx = du / (2x + 4).
Substituting the values in the integral, we have:
g(x) = ∫³√(x² + 4x) dx
= ∫(x² + 4x)^(1/3) dx
= ∫u^(1/3) * (1 / (2x + 4)) du
Now, the integral becomes:
g(x) = ∫u^(1/3) / (2x + 4) du
Integrating u^(1/3) with respect to u, we have:
g(x) = (3/(2x + 4)) * u^(4/3) + C
where C is the constant of integration.
Since g(5) = 7, we can use this information to find the constant C:
g(5) = (3/(2*5 + 4)) * (5² + 4*5)^(4/3) + C
7 = (3/14) * (45)^(4/3) + C
7 = (3/14) * 125 + C
7 = 26.7857 + C
C = 7 - 26.7857
C ≈ -19.7857
Now that we have the constant of integration, we can find g(1):
g(1) = (3/(2*1 + 4)) * (1² + 4*1)^(4/3) + C
= (3/6) * (5)^(4/3) - 19.7857
Using a calculator to find the approximate value of (5)^(4/3), we get:
g(1) ≈ (3/6) * 11.1803 - 19.7857
≈ 5.59015 - 19.7857
≈ -14.1956
Therefore, g(1) is approximately equal to -14.1956.
To find the value of g(1), we need to evaluate the antiderivative g at x = 1. Since g is an antiderivative of f, we can find g(x) using the indefinite integral of f(x).
First, let's find f(x) by substituting the given function definition into f(x) = cube root of (x² + 4x):
f(x) = cube root of (x² + 4x)
To find the indefinite integral of f(x), we integrate f(x) with respect to x:
∫ f(x) dx = ∫ cube root of (x² + 4x) dx
To solve this integral, we can use a substitution. Let's substitute u = x² + 4x, so du/dx = 2x + 4.
Substituting back into the integral, we have:
∫ cube root of (x² + 4x) dx = ∫ (1/3)*(u^(1/3))*(1/(2x + 4)) du
Simplifying further:
(1/2) * (1/3) * ∫ u^(1/3) du
(1/6) * ∫ u^(1/3) du
Using the power rule of integration, we integrate u^(1/3) as:
(1/6) * (3/4) * u^(4/3) + C'
(1/8) * u^(4/3) + C'
Substituting u back in, we have:
(1/8) * (x² + 4x)^(4/3) + C
Now, we can find g(x) by replacing C with a constant C that makes g(5) = 7:
g(x) = (1/8) * (x² + 4x)^(4/3) + C
To find C, we substitute x = 5 and g(5) = 7:
7 = (1/8) * (5² + 4*5)^(4/3) + C
7 = (1/8) * (25 + 20)^(4/3) + C
7 = (1/8) * (45)^(4/3) + C
Multiplying both sides by 8, we get:
56 = (45)^(4/3) + 8C
Subtracting (45)^(4/3) from both sides, we get:
56 - (45)^(4/3) = 8C
Now, we can solve for C:
C = (56 - (45)^(4/3)) / 8
After finding the value of C, we can find g(1) by substituting x = 1 into g(x):
g(1) = (1/8) * (1² + 4*1)^(4/3) + C
g(1) = (1/8) * (1 + 4)^(4/3) + C
g(1) = (1/8) * (5)^(4/3) + C
Substituting the value of C we found earlier:
g(1) = (1/8) * (5)^(4/3) + ((56 - (45)^(4/3)) / 8)
Now, you can evaluate this expression using a calculator or mathematical software to find the approximate value of g(1).