Given the function

Question

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

Find the slopes of the secant lines that pass though the points $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ , for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

Accepted Answer

Welcome back everyone. In this problem, consider the quadratic function F FX equals negative four X squared plus 32 X, calculate the slopes of the second lines between the points XF FX and four FF +44, X equals 3.53 0.93 0.99 and 3.999. For our answer choices A says they are 20.40 0.04 and 0.004 B 60.30 0.03 and 0.003 C 10, 0.50 0.05 and 0.005 and D 12, 0.80 0.06 and 0.003. Now, how can we figure out what the slopes of our lines are between our given points? Well, first, let's think about the slope. What do we know we recall that the slope is equal to the change in the Y direction divided by the change in the X direction for a line. So in this case, the slope for our C lines are going to be equal to the difference between the Y values. OK? In this case, we could say F of four minus F of X, OK, divided by the change in X values, which is four minus X. And that makes sense because remember we are trying to find the slopes of the C lines between the points XF of X. Let's just highlight that here XF FX and four FF four. Therefore, we, we know that our final Y value is always going to be FF four and our final X value is going to be four. So in that case to find the slopes of the C lines for different X points, we're going to substitute the values of X and F of X to eventually help us solve for the value of our slope. Now, before we do that, since FF four is a constant, let's go ahead and solve. OK. So at X being equal to four, then we know that F of four is going to be equal to negative four, multiplied by four squared minus 32 multiplied by 44 squared is 16, negative four, multiplied by 16 equals negative 64. OK? And no, well, this should be plus here. My apologies, 32 multiplied by four equals 128 negative 64 plus 128 equals 64. So FF four is 64. No, we need to go ahead and to find the values of our function F for our varying X values. So let's start with X being equal to 3.5. Now add X equals 3.5. OK? Then we know that F of 3.5 is going to be equal to negative four, multiplied by 3.5 squared plus 32 multiplied by 3.5. And when we work that out, we should get F of 3.5 to be equal to 63. That makes sense because it's less than FF four. Now that we have F of 3.5 this tells us then that the slope, OK? At X equals 3.5 is going to be equal to 64 minus 63 divided by four minus 3.5. That's going to be one divided by 0.5 which equals two. So at X equals 3.5 or slope is two. Next, let's move on to X being equal to 3.9. Now, at XE cost 3.9 we're going to find Fox 3.9. That's going to be negative four, multiplied by 3.9 squared plus 32 multiplied by 3.9. That equals 63.96. OK? Now, when we go ahead and solve for our slope, then slope is going to be equal to 64 minus 63.96 divided by four minus 3.9. And when we work that out, we get the slope to be equal to 0.4. Next, let's move on. Let's move on to X being equal to 3.99. Now, at X being equal to 3.99 we're gonna find F of 3.99 that equals negative four, multiplied by 3.99 squared plus 32 multiplied by 3.99 which is going to be equal to 63.9996. OK. And no, that means the slope is going to be equal to 64 minus 63.9996. OK. All divided by four minus 3.99. And when we work that out, we get it to be 0.04. Let's do our final one. I think I'll need some space here. So let me just try to make some space on the left. OK? So now let me put it in blue just to show what I'm talking about next. Let's look at when X equals 3.999 the next is 3.999 or function is going to be negative four multiplied by 3.999 squared plus 32 multiplied by 3.999. When we were at o we get it to be equal to 63.999996. Now that we have that, that tells us then that the slope OK, is going to be equal to 64 minus 63.999996. OK. All divided by four minus 3.999. And when we write that out, we get the slope to be 0.004. Thus, that tells us then that our slopes are 20.40 0.04 and 0.004. If we look back at our answer choices that tells us that a 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>
Find the slopes of the secant lines that pass though the points $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ , for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

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