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‘₯, 𝑦) on the curve the slope of the tangent at that point is βˆ’3π‘₯⁡ the ninth root of π‘₯⁸.

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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‘₯, 𝑦 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, we’re given the slope function or d𝑦 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, we’ll need to integrate this function. However, before doing this, we will try to simplify the expression for d𝑦 by dπ‘₯. We begin by recalling one of our exponent laws. This state that the 𝑛th root of π‘₯ is equal to π‘₯ to the power of one over 𝑛. 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‘Ž raised to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž multiplied by 𝑏. 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‘Ž multiplied by π‘₯ to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž plus 𝑏. 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 d𝑦 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. 𝑦 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 𝑛 with respect to π‘₯ is equal to π‘₯ to the power of 𝑛 plus one divided by 𝑛 plus one plus our constant of integration 𝐢. Adding one or nine-ninths to our power gives us sixty-two ninths. And 𝑦 is therefore equal to negative three π‘₯ to the power of 62 over nine divided by 62 over nine plus 𝐢. This in turn simplifies to 𝑦 is equal to negative 27π‘₯ to the power of 62 over nine all divided by 62 plus the constant of integration 𝐢.

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

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