Given the function

Question

Given the function $f\left(x\right)=-16x^2+64x$ , complete the following.

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ as $x$ approaches $2$ .

Accepted Answer

Welcome back, everyone consider the quadratic function F of X equals negative for X squared plus 32 X, calculate the slopes of the C current lines between the points XF of X and four F of four for X equals 3.53 0.93 0.99 and 3.999. Then make a presumption on the value of the limit of the slopes of the secret lines between XF of X and four FF four. As X gets closer to four. Here we have a graph of F of X. And for our answer choices, A says the slopes are 12 0.80 0.06 and 0.003 where the limit is zero B 60.30 0.030 0.003. The limit is six, C 10, 0.50 0.050 0.005. And the limit is 10 and the D 20.40 0.040 0.004. And the limit is zero. Now, before we try to figure out the value of the limit of the slopes of the second lines. OK. Let's first calculate the slopes of the second lines and then see where we can go from there. Now, what do we know about the slope of a line? Recall the, the slope of a line is going to be equal to its change in the Y direction divided by the change in the X direction. So for our slope here, based on our graph, if we call it M it's going to be equal to the difference in value between the or sorry, the difference between the Y values in this case would be going from F FX to FF four. So that's really FF four minus F of X. OK. Divided by the difference in the X values which would go from X to four. So that's four minus X. So to find the slope for each point on our graph, all we need to do is to substitute the X value to find F FX and use it in the slope equation to solve. But before we continue notice that FF four is going to be in all of the slope equations. It's a constant. So let's try to find it first. Before we move on now to find F of four, we're going to substitute four into our function. So that's going to be negative four. OK? Multiplied by four squared plus 32 multiplied by +44 squared is 16, negative four, multiplied by 16 equals negative 64. And when we add that to 32 multiplied by four, which is 128 F of four equals 64. Ok. So that is going to be our function F four. Now that we have that we can go ahead and start to solve for our slopes. Let's start with X being equal to 3.5. No add X equals 3.5. Yeah, X equals 3.5. Then first to find FF 3.5 we can substitute 3.5 into our function. So that's gonna be negative four, multiplied by 3.5 squared plus 32 multiplied by 3.5. And when we go ahead and solve here, it's going to be equal to 63. In that case, then that means our slope is going to be equal to FF four, which is 64 minus F of 3.5 63 divided by four minus 3.5. When we write, thought we get our slope to be equal to two. Next, let's find it for X being equal to 3.9. OK? Now at X equals 3.9. OK. Then F of 3.9 is going to be equal the negative four multiplied by 3.9 squared plus 32 multiplied by 3.9 which is going to be equal to 63.96. Therefore, that tells us our slope. M, let me write it over here. That means M equals 64 minus 63.96 divided by four minus 3.9 which is going to be equal. OK. Let me just fix this here. It's going to be equal to 0.4. Thus that is our slope at X equals 3.9. Next add X equals 3.99. OK. Then ff 3.99 cause negative four multiplied by 3.99 squared plus 32 multiplied by 3.99. OK. I know when we work that out, we get F of 3.99 to be 63.9996. OK. Which means then that our slope M equals 64 minus 63.9996 divided by four minus 3.99 which is going to be equal to 0.04. Finally let's find our slope at X equals 3.999. OK. No, if a 3.999 is going to be equal. Ok. Sorry, it's gonna be equal to negative four. OK. Multiplied by 3.999 squared plus 32 multiplied by 3.999 which is gonna be equal to 63.999996. OK. Which means then that the slope M it's gonna be equal to 64 minus 63.999996 divided by one minus 3.999 which when we work, that thought we get the slope to be 0.004. So here we have the values of our slopes for sea lands at the different points. Now, the question is, what is the limit? What is the value of the limit as X gets closer to four? Well, we can look at our values starting with 3.9. OK. At 3.9 F sorry, our slope is 0.4 at 3.99 the slope is 0.004. And at 3.999 the slope is 0.004. What do we notice? Well, now we can tell by by looking that as X approaches four, the slopes of the secret lines that pass through FX F FX and four FF four get closer to zero. Hence, a good presumption is that the limit is zero. And that makes sense. OK. That makes sense when we think about the limit. So then we can say that let me put it in back here as X approaches four E, the slope approaches zero. And again, we know that makes sense because if we go back to our diagram, OK, if we think about it as this is this, so this would be the slope of our line at a point. Notice that as it approaches four, our slope gets straighter. OK. And when we have a horizontal slope, then its gradient is equal to zero, its value is equal to zero. So it makes sense that the slope gets closer to zero. If we look back at our answer choices that tells us that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Given the function $f\left(x\right)=-16x^2+64x$ , complete the following. <IMAGE>
Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ as $x$ approaches $2$ .

Key Concepts

Secant Lines

Limits

Derivatives

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