Question Video: Finding the Equation of a Curve given the Expression of the Slope of Tangent Using Integration

Video Transcript

Find the equation of a curve which passes through the point zero, zero, and for each point 𝑎𝑏 on the curve, the slope of the tangent at that point is negative three 𝑥 to the power of five the ninth root of 𝑥 to the power of eight.

If we know that the slope function or 𝑑𝑦 𝑑𝑥 is equal to negative three 𝑥 to the power of five the ninth root of 𝑥 to the power of eight, then we know that what we can actually do to find 𝑦 is first of all integrate this.

However, what we’re actually going to do before we integrate the expression is rewrite it to make it a more simple process. And to enable us to do that, what we’re actually gonna use is an exponent rule. And the exponent rule we’re gonna use is 𝑎 to the power of one over 𝑏 equals the 𝑏th root of 𝑎. So, therefore, we can actually use that now to rewrite our expression.

So when we apply the exponent rule, we get negative three 𝑥 to the power of five and then multiplied by 𝑥 to the power of eight over nine. And that’s because our ninth root became one over nine. And then we had 𝑥 to the power of eight.

But should we integrate now? I think actually we can simplify even further. So let’s do that. So to actually simplify it further, we can use another exponent rule, which is that 𝑎 to the power of 𝑏 multiplied by 𝑎 to the power of 𝑐 equals 𝑎 to the power of 𝑏 plus 𝑐. So therefore, we can actually add the exponents. And when we apply this rule, we get negative three 𝑥 to the power of 53 over nine. And I’ve shown you how we got that because five is equal to 45 over nine and 45 over nine plus eight over nine gives us 53 over nine.

Okay, great! So now let’s integrate our expression. So when we integrate the expression, we get negative three 𝑥 to the power of 62 over nine over 62 over nine plus 𝑐. And just to explain how we got there, we got negative three 𝑥 to the power of 53 over nine. And then we added nine over nine, or one. So we added one to the exponent. And then we divided by the new exponents. So therefore, we know that if we’re gonna simplify, we get negative three 𝑥 to the power of 62 over nine multiplied by nine over 62 plus 𝑐. And we get that because if we divide by a fraction, it’s actually the same as multiplying by the reciprocal of that fraction.

So now from that, what we can say is that 𝑦 is equal to negative 27𝑥 to the power of 62 over nine over 62 then plus 𝑐. So great! We’ve actually found the equation of the curve, or have we? Well, we’re not quite there because we’ve still got plus 𝑐. So we still got our constant of integration on the end. So we need to find out what this will be.

Well, to help us do this, we actually have a piece of information because we know one of the points on our curve. We know that the curve passes through the point zero, zero. So therefore, we know we can find 𝑐 by substituting in the coordinates zero, zero. So the 𝑥 is equal to zero. And 𝑥 is equal to zero coordinates. And that will allow us to find our 𝑐. So this gives us that zero equals negative 27 multiplied by zero to the power of 62 over nine all over 62 plus 𝑐, which just takes us to zero equals zero plus 𝑐.

So therefore, 𝑐 must just be equal to zero. So we can use that. And we can now write that our final equation of the curve which passes through the point zero, zero and has a slope function of negative three 𝑥 to the power of five the ninth root of 𝑥 to the power of eight is gonna be 𝑦 is equal to negative 27𝑥 to the power of 62 over nine over 62.