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Math  /  Algebra

QuestionFind all points on the graph of y3−27y=x2−90y^{3}-27 y=x^{2}-90 at which the tangent line is vertical. (Order your answers from smallest to largest xx, then from smallest to largest yy.) (x,y)=(□)(x,y)=(□)(x,y)=(□)(x,y)=(□)\begin{array}{l} (x, y)=(\square) \\ (x, y)=(\square) \\ (x, y)=(\square) \\ (x, y)=(\square) \end{array}

Studdy Solution

STEP 1

1. A vertical tangent line occurs when the derivative of the function with respect to x x is undefined.
2. The implicit differentiation technique will be used since the equation is not explicitly solved for y y .

STEP 2

1. Differentiate the given equation implicitly with respect to x x .
2. Set the derivative of x x with respect to y y equal to zero to find vertical tangents.
3. Solve the resulting equation to find potential y y -values.
4. Substitute y y -values back into the original equation to find corresponding x x -values.
5. List the points in the specified order.

STEP 3

Differentiate the given equation y3−27y=x2−90 y^3 - 27y = x^2 - 90 implicitly with respect to x x :
ddx(y3−27y)=ddx(x2−90)\frac{d}{dx}(y^3 - 27y) = \frac{d}{dx}(x^2 - 90)
Using the chain rule, we have:
3y2dydx−27dydx=2x3y^2 \frac{dy}{dx} - 27 \frac{dy}{dx} = 2x

STEP 4

For the tangent line to be vertical, the derivative dydx\frac{dy}{dx} must be undefined, which occurs when the denominator of dydx\frac{dy}{dx} is zero. Rearrange the differentiated equation:
dydx=2x3y2−27\frac{dy}{dx} = \frac{2x}{3y^2 - 27}
Set the denominator equal to zero:
3y2−27=03y^2 - 27 = 0

STEP 5

Solve the equation 3y2−27=0 3y^2 - 27 = 0 :
3y2=273y^2 = 27 y2=9y^2 = 9 y=3ory=−3y = 3 \quad \text{or} \quad y = -3

STEP 6

Substitute y=3 y = 3 and y=−3 y = -3 back into the original equation to find corresponding x x -values.
For y=3 y = 3 :
33−27×3=x2−903^3 - 27 \times 3 = x^2 - 90 27−81=x2−9027 - 81 = x^2 - 90 x2=90−54x^2 = 90 - 54 x2=36x^2 = 36 x=6orx=−6x = 6 \quad \text{or} \quad x = -6
For y=−3 y = -3 :
(−3)3−27×(−3)=x2−90(-3)^3 - 27 \times (-3) = x^2 - 90 −27+81=x2−90-27 + 81 = x^2 - 90 x2=90−54x^2 = 90 - 54 x2=36x^2 = 36 x=6orx=−6x = 6 \quad \text{or} \quad x = -6

STEP 7

List the points in the specified order (smallest to largest x x , then smallest to largest y y ):
1. (x,y)=(−6,−3) (x, y) = (-6, -3)
2. (x,y)=(−6,3) (x, y) = (-6, 3)
3. (x,y)=(6,−3) (x, y) = (6, -3)
4. (x,y)=(6,3) (x, y) = (6, 3)
The points are: (x,y)=(−6,−3)(x, y) = (-6, -3) (x,y)=(−6,3)(x, y) = (-6, 3) (x,y)=(6,−3)(x, y) = (6, -3) (x,y)=(6,3)(x, y) = (6, 3)

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