Video Transcript
Find the equation of a curve that
passes through the point zero, zero and is such that for each point 𝑥, 𝑦 on the
curve the slope of the tangent at that point is negative three 𝑥 to the power of
five multiplied by the ninth root of 𝑥 to the power of eight.
In this question, we’re given the
slope function or d𝑦 by d𝑥, which is equal to negative three 𝑥 to the power of
five multiplied by the ninth root of 𝑥 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 d𝑥. We begin by recalling one of our
exponent laws. This state that the 𝑛th root of 𝑥
is equal to 𝑥 to the power of one over 𝑛. We can therefore rewrite the second
term in our expression as 𝑥 to the power of eight raised to the power of
one-ninth.
Next, we recall the power rule of
exponents. This states that 𝑥 to the power of
𝑎 raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. Multiplying eight by one-ninth, our
expression simplifies to negative three 𝑥 to the power of five multiplied by 𝑥 to
the power of eight-ninths. Finally, we recall that 𝑥 to the
power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎
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 d𝑥 simplifies to negative three 𝑥 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 𝑥 to the power of fifty-three ninths with respect to 𝑥. The power rule of integration
states that the integral of 𝑥 to the power of 𝑛 with respect to 𝑥 is equal to 𝑥
to the power of 𝑛 plus one divided by 𝑛 plus one plus our constant of integration
𝐶. Adding one or nine-ninths to our
power gives us sixty-two ninths. And 𝑦 is therefore equal to
negative three 𝑥 to the power of 62 over nine divided by 62 over nine plus 𝐶. This in turn simplifies to 𝑦 is
equal to negative 27𝑥 to the power of 62 over nine all divided by 62 plus the
constant of integration 𝐶.
Our next step is to find the value
of the constant of integration 𝐶. We are told that the curve passes
through the origin, the point with coordinates zero, zero. Substituting 𝑥 is equal to zero
and 𝑦 is equal to zero into our equation, we see that 𝐶 is also equal to zero. We can therefore conclude that the
equation of the curve is 𝑦 is equal to negative 27𝑥 to the power of 62 over nine
divided by 62.
