Video: Finding the Unknowns in a Rational Function given Its Value and the Value of Its First Derivative at a Point

Suppose that 𝑓(π‘₯) = (π‘₯Β² + π‘Žπ‘₯ + 𝑏)/(π‘₯Β² βˆ’ 7π‘₯ + 4). Given that 𝑓(0) = 1 and 𝑓′(0) = 4, find π‘Ž and 𝑏.

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Video Transcript

Suppose that 𝑓 of π‘₯ is equal to π‘₯ squared plus π‘Žπ‘₯ plus 𝑏 all over π‘₯ squared minus seven π‘₯ plus four. Given that 𝑓 of zero is equal to one and 𝑓 prime of zero is equal to four, find π‘Ž and 𝑏.

Our first step in this question can be to substitute π‘₯ equals zero into 𝑓 of π‘₯. Since we’re given that 𝑓 of zero is equal to one. We obtain that 𝑓 of zero is equal to zero squared plus π‘Ž times zero plus 𝑏 all over zero squared minus seven times zero plus four. Now, all of these terms will go to zero apart from 𝑏 and four. We’re left with 𝑓 of zero is equal to 𝑏 over four. Next, we use the fact that the question has told us that 𝑓 of zero is equal to one. And so, we can set this equal to one. From this, we find that 𝑏 is equal to four. Next, we can use the fact that 𝑓 prime of zero is equal to four. However, first of all, we must find 𝑓 prime of π‘₯. In order to do this, we need to differentiate 𝑓. Since 𝑓 is a rational function, we can use the quotient rule in order to find its derivative.

The quotient rule tells us that 𝑒 over 𝑣 prime is equal to 𝑣 times 𝑒 prime minus 𝑒 times 𝑣 prime all over 𝑣 squared. Setting our function 𝑓 of π‘₯ equal to 𝑒 over 𝑣, we obtain that 𝑒 is equal to π‘₯ squared plus π‘Žπ‘₯ plus 𝑏. And 𝑣 is equal to π‘₯ squared minus seven π‘₯ plus four. We can find 𝑒 prime and 𝑣 prime by differentiating these two functions. Giving us that 𝑒 prime is equal to two π‘₯ plus π‘Ž and 𝑣 prime is equal to two π‘₯ minus seven. Now, we can substitute these into the quotient rule. We obtain that 𝑓 prime of π‘₯ is equal to π‘₯ squared minus seven π‘₯ plus four multiplied by two π‘₯ plus π‘Ž minus π‘₯ squared plus π‘Žπ‘₯ plus 𝑏 multiplied by two π‘₯ minus seven all over π‘₯ squared minus seven π‘₯ plus four all squared.

Now, we could simplify 𝑓 prime of π‘₯ at this point. However, we’re going to be substituting in π‘₯ is equal to zero. And so, a lot of these terms would just disappear. Let’s simply substitute π‘₯ equals zero in here. We obtain this. However, a lot of the terms will vanish to zero, which leaves us with four π‘Ž plus seven 𝑏 all over 16. Now, we have found that 𝑏 is equal to four earlier. And so, we can substitute this in, giving us four π‘Ž plus 28 all over 16.

Since the question has told us that 𝑓 prime of zero is equal to four, we can set this equal to four. Then, we simply rearrange this in order to solve for π‘Ž. Now, we obtain our solution that π‘Ž is equal to nine. We’ve now found the values of both π‘Ž and 𝑏, which completes the solution to this question.

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