Find two functions defined implicitly by the given relations:
(a) 3x² - y² = 25
(b) y² = x
Isolate y on the left-hand side, and solve for y.
I'll do the first one.
(a) 3x²-y²=25
-y²=25-3x²
y²=3x²-25
y = √(3x²-25)
(a) To find two functions defined implicitly by the relation 3x² - y² = 25, we can solve for y in terms of x.
Step 1: Start with the given relation:
3x² - y² = 25
Step 2: Move the y-term to the left side:
-y² = 25 - 3x²
Step 3: Divide both sides by -1 to make y² positive:
y² = -25 + 3x²
Step 4: Take the square root of both sides:
y = ±√(-25 + 3x²)
So, two functions defined implicitly by the relation 3x² - y² = 25 are:
f(x) = √(-25 + 3x²)
g(x) = -√(-25 + 3x²)
(b) To find two functions defined implicitly by the relation y² = x, we can solve for y in terms of x.
Step 1: Start with the given relation:
y² = x
Step 2: Take the square root of both sides:
y = ±√x
So, two functions defined implicitly by the relation y² = x are:
f(x) = √x
g(x) = -√x
To find two functions defined implicitly by the given relations, we can express one variable in terms of the other variable and differentiate both sides of the equation to find the slope. Then, we can use this information to find the equation of the tangent line and solve for the other variable.
Let's start with the first relation:
(a) 3x² - y² = 25
To find a function defined implicitly in terms of x, we can express y in terms of x. Rearrange the equation as follows:
3x² - y² = 25
y² = 3x² - 25
y = ±√(3x² - 25)
Now, let's find the slope of the function by differentiating both sides of the equation with respect to x:
dy/dx = ±(1/2) * (3x² - 25)^(-1/2) * (6x)
dy/dx = ±(3x)/(2√(3x² - 25))
To find the equation of the tangent line, we need a point on the curve. Let's choose a point (x₀, y₀) on the curve. We can substitute this point into the equation and the slope equation and solve the system of equations to find the values of x₀ and y₀.
Once we have x₀ and y₀, we can use the point-slope form of a line to find the equation of the tangent line:
(y - y₀) = m(x - x₀)
Similarly, let's move on to the second relation:
(b) y² = x
To find a function defined implicitly in terms of y, we can express x in terms of y. Rearrange the equation as follows:
y² = x
x = y²
Since x is already expressed in terms of y, we don't need to differentiate this equation to find the slope. We can directly use the equation to find the equation of the tangent line.
Again, choose a point (x₀, y₀) on the curve and substitute it into the equation to find the values of x₀ and y₀.
Once we have x₀ and y₀, we can use the point-slope form of a line to find the equation of the tangent line:
(y - y₀) = m(x - x₀)
Remember, the equations of the tangent lines represent functions implicitly defined by the given relations.