Suppose f(x) = 4x^5-(1/x^4)
Find the slope of the tangent line to the graph of y = f^–1(x) at (3, 1).
You get:
1/24
1/16
00062
1620
972
Got this one. Its 1/24 correct?
Its 1/24, the slope of the tangent line to the graph y=f^-1(x) at (3,1) is the reciprocal of the slope of the tangent line to the graph y=f(x) at (1,3).
So f'(x)=20x^4+4/x^5 so f'(1)=20+4=24
So the slope of the tangent is 1/24
Well, finding the slope of a tangent line can sometimes be a slippery slope, but fear not, my friend! Let's tackle this problem together.
First, we need to find the derivative of the inverse function. Now, without going into any complex mathematical acrobatics, let's just proceed carefully. We know that the inverse function is the reflection of the original function in the line y = x. So, we can switch the x and y variables in our function to make things a bit easier.
Now, we have x = 4y^5 - (1/y^4). Still with me? Good. Now, if we differentiate both sides of this equation with respect to x, we can find the derivative of the inverse function.
To do that, we use implicit differentiation. Taking the derivative of both sides, we get 1 = 20y^4(dy/dx) + 4/y^5.
Now we just need to solve for dy/dx. Rearranging the equation, we get dy/dx = (1 - 4/y^5) / (20y^4).
Fantastic! Now we have dy/dx, which represents the slope of the tangent line to the inverse function. All that's left to do is substitute the values of x = 3 and y = 1 into this equation to get our answer.
So, plugging in the values, we have dy/dx = (1 - 4/1^5) / (20(1)^4) = (1 - 4) / (20) = -3/20.
Therefore, the slope of the tangent line to the graph of y = f^–1(x) at (3, 1) is -3/20.
Remember, math can sometimes be a circus of complexity, but I'm here to clown around and make it a bit more entertaining!
To find the slope of the tangent line to the graph of y = f^–1(x) at (3, 1), we can use the fact that the slope of the tangent line to the graph of y = f(x) at a point (a, b) is given by the derivative of f(x) evaluated at that point, which is f'(a).
Step 1: Find the derivative of f(x):
f(x) = 4x^5 - (1/x^4)
To find the derivative, we can use the power rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1).
Applying the power rule to each term of f(x), we get:
f'(x) = 20x^4 + 4/x^5
Step 2: Evaluate the derivative at x = 1/f(3):
We need to find f'(1/f(3)).
First, we need to find f(3):
f(3) = 4(3)^5 - (1/(3)^4)
= 972 - 1/81
= 971/81
Now, we can find x = 1/f(3):
x = 1/(971/81)
= 81/971
So, we need to evaluate f'(x) at x = 81/971:
f'(81/971) = 20(81/971)^4 + 4/(81/971)^5
= 20(81^4/971^4) + 4/(81^5/971^5)
= 20(531441/91980016881041) + 4/(531441/91980016881041)^5
≈ 20(0.000000005773) + 4/(0.000000005773)^5
≈ 0.0000001155 + 4/1.671e-47
≈ 0.0000001155 + 2.393e47
Therefore, the slope of the tangent line to the graph of y = f^–1(x) at (3, 1) is approximately:
0.0000001155 + 2.393e47 ≈ 2.393e47.
So, the correct answer is 1620 (option D).
To find the slope of the tangent line to the graph of y = f^–1(x) at a given point, we'll follow these steps:
Step 1: Find the inverse function, f^–1(x), of the original function, f(x).
Step 2: Differentiate the inverse function, f^–1(x), to find its derivative.
Step 3: Substitute the x-coordinate of the point of interest into the derivative obtained in Step 2.
Let's go through these steps one by one:
Step 1: Finding the inverse function, f^–1(x), of f(x):
To find the inverse function, you need to swap the roles of x and y and then solve for y.
So, let's start by rewriting the original function as an equation:
x = 4y^5 - (1/y^4)
Now, we can solve this equation for y. First, let's multiply both sides by y^4 to eliminate the fraction:
xy^4 = 4y^9 - 1
Rearranging the equation, we get:
4y^9 - xy^4 - 1 = 0
Now, we can treat this as a polynomial equation and solve for y. However, finding the exact inverse function can be quite laborious. Instead, we can use numerical methods or approximation techniques to find the inverse function.
For the purposes of this explanation, let's assume that we have obtained the inverse function, f^–1(x), as:
f^–1(x) ≈ 1/3
Step 2: Differentiating the inverse function, f^–1(x), to find its derivative:
Since f^–1(x) is a constant function here, its derivative will be zero. Therefore, f^–1'(x) = 0.
Step 3: Substitute the x-coordinate of the point (3, 1) into the derivative obtained in Step 2:
Since f^–1'(x) = 0, plugging in any x-value will give us the same result of zero. So, regardless of the point we choose, the slope of the tangent line to the graph of y = f^–1(x) will be zero.
Therefore, none of the options provided (1/24, 1/16, 0.0062, 1620, or 972) are correct.
In conclusion, the slope of the tangent line to the graph of y = f^–1(x) at (3, 1) is 0.