The curve with the equation y^2=5x^4-x^2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (-1,2).
y^2=5x^4-x^2
2y dy/dx = 20x^3 - 2x
dy/dx = (20x^3 - 2x)/(2y)
= (10x^3 - x)/y
at (-1,2)
dy/dx = (-10+1)/2
= -9/2
so the tangent equation is 9x + 2y = c
with (-1,2) lying on it, so
-9 + 4 = c = -5
the tangent equation is 9x + 2y = -5
check:
http://www.wolframalpha.com/input/?i=y%5E2%3D5x%5E4-x%5E2+%2C+9x+%2B+2y+%3D+-5+%2C+from+-2+to+0
nope that is incorrect
the tangent equation is -9/2x-5/2
First of all -9/2x-5/2 is NOT an equation
if you meant y = -(9/2)x - 5/2
then by golly, that is what my equation is
9x + 2y = -5
2y = -9x - 5
divide each term by 2
y = (-9/2)x - 5/2
My answer is correct.
well the online homework system didn't take it as a complete equation because the y= part was already outside of the answer box
well, there you go, how about that
It worked for mine, thanks Reiny.
To find the equation of the tangent line to the curve y^2 = 5x^4 - x^2 at the point (-1,2), we can use the concept of differentiation. Differentiation allows us to find the slope of the curve at any given point, which can then be used to determine the equation of a tangent line.
Step 1: Differentiate both sides of the equation y^2 = 5x^4 - x^2 with respect to x.
Let's differentiate y^2 with the chain rule: d/dx (y^2) = 2y * (dy/dx).
On the right side, the first term becomes 20x^3 due to the power rule, and the second term becomes -2x due to the power rule.
So we have: 2y * (dy/dx) = 20x^3 - 2x.
Step 2: Now, substitute the coordinates of the given point (-1,2) into the equation obtained from Step 1.
Replacing x with -1 and y with 2, we get: 2* (dy/dx) = 20*(-1)^3 - 2*(-1).
Simplifying gives: 2 * (dy/dx) = -20 + 2.
Therefore: 2 * (dy/dx) = -18.
Step 3: Solve for (dy/dx) which represents the slope of the tangent line at (-1,2).
Dividing both sides of the equation by 2, we find: (dy/dx) = -9.
So, the slope of the tangent line at (-1,2) is -9.
Step 4: Use the point-slope form (y - y1) = m(x - x1), where m represents the slope and (x1, y1) the given point, to find the equation of the tangent line.
Plugging in the values for m = -9 and (x1, y1) = (-1,2), we get:
(y - 2) = -9(x - (-1)).
Simplifying further gives:
y - 2 = -9x - 9.
Finally, rearrange the equation into standard form to obtain the equation of the tangent line:
9x + y = -7.
Therefore, the equation of the tangent line to the curve y^2 = 5x^4 - x^2 at the point (-1,2) is 9x + y = -7.