Question Video: Finding the Equation of a Curve Given the Slope of the Tangent and a Point that Lies on the Curve | Nagwa Question Video: Finding the Equation of a Curve Given the Slope of the Tangent and a Point that Lies on the Curve | Nagwa

# Question Video: Finding the Equation of a Curve Given the Slope of the Tangent and a Point that Lies on the Curve Mathematics • Third Year of Secondary School

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Find the equation of a curve that passes through the point (0, 0) and is such that for each point (π₯, π¦) on the curve the slope of the tangent at that point is β3π₯β΅ the ninth root of π₯βΈ.

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### Video Transcript

Find the equation of a curve that passes through the point zero, zero and is such that for each point π₯, π¦ on the curve the slope of the tangent at that point is negative three π₯ to the power of five multiplied by the ninth root of π₯ to the power of eight.

In this question, weβre given the slope function or dπ¦ by dπ₯, which is equal to negative three π₯ to the power of five multiplied by the ninth root of π₯ to the power of eight. And we know that in order to find the equation of the curve, weβll need to integrate this function. However, before doing this, we will try to simplify the expression for dπ¦ by dπ₯. We begin by recalling one of our exponent laws. This state that the πth root of π₯ is equal to π₯ to the power of one over π. We can therefore rewrite the second term in our expression as π₯ to the power of eight raised to the power of one-ninth.

Next, we recall the power rule of exponents. This states that π₯ to the power of π raised to the power of π is equal to π₯ to the power of π multiplied by π. Multiplying eight by one-ninth, our expression simplifies to negative three π₯ to the power of five multiplied by π₯ to the power of eight-ninths. Finally, we recall that π₯ to the power of π multiplied by π₯ to the power of π is equal to π₯ to the power of π plus π. We need to add the exponents five and eight-ninths. Since five can be rewritten as forty-five ninths, we have forty-five ninths plus eight-ninths. This is equal to fifty-three ninths.

Our expression for the slope function dπ¦ by dπ₯ simplifies to negative three π₯ to the power of fifty-three ninths. We are now in a position to integrate this expression to find the equation of the curve. π¦ is equal to the integral of negative three π₯ to the power of fifty-three ninths with respect to π₯. The power rule of integration states that the integral of π₯ to the power of π with respect to π₯ is equal to π₯ to the power of π plus one divided by π plus one plus our constant of integration πΆ. Adding one or nine-ninths to our power gives us sixty-two ninths. And π¦ is therefore equal to negative three π₯ to the power of 62 over nine divided by 62 over nine plus πΆ. This in turn simplifies to π¦ is equal to negative 27π₯ to the power of 62 over nine all divided by 62 plus the constant of integration πΆ.

Our next step is to find the value of the constant of integration πΆ. We are told that the curve passes through the origin, the point with coordinates zero, zero. Substituting π₯ is equal to zero and π¦ is equal to zero into our equation, we see that πΆ is also equal to zero. We can therefore conclude that the equation of the curve is π¦ is equal to negative 27π₯ to the power of 62 over nine divided by 62.

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