Show a graphical method of approximating the instantaneous rate of change at x = 3 for the function ƒ(x) = -x2 + 4x + 1 using secants. Show two graphical approximations.

Question

Show a graphical method of approximating the instantaneous rate of change at x = 3 for the function ƒ(x) = -x2 + 4x + 1 using secants. Show two graphical approximations.

I had the same question but instructed to find it using a numerical method.
No problem with that, but Im not sure how to calculate graphically.

Answer 1

just draw several position vectors v_{_k}, for i=1..n, getting closer to the vector at the given point, r. For each v_{_k}, draw v_{_k}-r_{. These serve the same purpose as secants in the usual x-y plots illustrating the approach to a tangent.}

Pick up any good book on vector analysis and you will find such illustrations. Also, as always, google is your friend.

Answer 2

Shoot. I see that you were in fact just talking about an x-y plot. Pick any point P on the curve, and several nearby points, q_{_k}. Draw the secants to the various points. You can see that as q_{_k} approaches P, the secant is ever more nearly a tangent.

I'm sure your textbook has this very illustration, maybe for some other f(x).

Answer 3

To approximate the instantaneous rate of change at x = 3 graphically using secants, you can follow these steps:

Step 1: Plot the function ƒ(x) = -x^2 + 4x + 1 on a graph.

Step 2: Draw a secant line between two points on the graph that approach x = 3. Choose two x-values, one on the left side of x = 3 and another on the right side. For example, you can choose x = 2 and x = 4.

Step 3: Calculate the slope of the secant line. Use the formula: slope = (ƒ(x2) - ƒ(x1)) / (x2 - x1). Plug in the values to find the slope.

Step 4: Repeat steps 2 and 3, but choose different x-values to get a second approximation.

Here's an example of how to do this:

Step 1: Plot the function ƒ(x) = -x^2 + 4x + 1 on a graph:

Function graph:
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___________|___________
|

Step 2: Draw a secant line between two points on the graph approaching x = 3. Let's choose x = 2 and x = 4:

Function graph with secant line (approximation 1):
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_______/___|__________
/ |

Step 3: Calculate the slope of the secant line:
slope = (ƒ(x2) - ƒ(x1)) / (x2 - x1)
= (ƒ(4) - ƒ(2)) / (4 - 2)
= (-(4^2) + 4(4) + 1 - (-(2^2) + 4(2) + 1)) / (4 - 2)
= (-(16) + 16 + 1 - (-(4) + 8 + 1)) / (4 - 2)
= (1 - (-5)) / (4 - 2)
= 6 / 2
= 3

This slope of 3 represents the first approximation of the instantaneous rate of change at x = 3.

Function graph with secant line (approximation 2):
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| /
___________|____/
|

Step 4: Repeat steps 2 and 3 with different x-values to get a second approximation. Let's choose x = 2.5 and x = 3.5:

Function graph with secant line (approximation 2):
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| /
___________|___/____
|

Calculate the slope the same way as in step 3. This will give you the second approximation of the instantaneous rate of change at x = 3.

Remember, as you choose x-values closer to x = 3, the secant lines will approximate the tangent line and provide a better approximation of the instantaneous rate of change at x = 3.

Show a graphical method of approximating the instantaneous rate of change at x = 3 for the function ƒ(x) = -x2 + 4x + 1 using secants. Show two graphical approximations.

just draw several position vectors vk, for i=1..n, getting closer to the vector at the given point, r. For each vk, draw vk-r. These serve the same purpose as secants in the usual x-y plots illustrating the approach to a tangent.

Shoot. I see that you were in fact just talking about an x-y plot. Pick any point P on the curve, and several nearby points, qk. Draw the secants to the various points. You can see that as qk approaches P, the secant is ever more nearly a tangent.

To approximate the instantaneous rate of change at x = 3 graphically using secants, you can follow these steps:

just draw several position vectors v_{_k}, for i=1..n, getting closer to the vector at the given point, r. For each v_{_k}, draw v_{_k}-r_{. These serve the same purpose as secants in the usual x-y plots illustrating the approach to a tangent.}

Shoot. I see that you were in fact just talking about an x-y plot. Pick any point P on the curve, and several nearby points, q_{_k}. Draw the secants to the various points. You can see that as q_{_k} approaches P, the secant is ever more nearly a tangent.