For each function ƒ and interval [a, b], a graph of ...

Question

For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.

a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .

b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

ƒ(x) = x⁵/16 ; [-2, 2]

Accepted Answer

Welcome back everyone. to another video. Function G is graphed on the interval from -3 to 3 inclusive. A second line that passes through the curve G at X equals -3, and X equals 3 is shown below. Find the value or values of C in the interval from -3 to 3 inclusive, for which the equation F of B minus F of A divided by B minus A equals F C holds true. We're given the graph of the function. We're given the expression of the function, and essentially what we're going to do is simply define the end points of the interval. So our initial point is A equals -3 in the given interval, B is 3, and we're going to apply the mean value theorem right because we have a continuous and differentiable function. So let's identify the derivative of G of X. We want to differentiate -19. Ax cubed So we can factor out the constant, right, and essentially applying the power rule we're going to get -19th multiplied by 3 X2, and this can be simplified to negative X2 divided by 3. Now what we're going to do is simply take this derivative and equate it to f of B minus F of A divided by b minus A. If we use that expression, this gives us F at 0.3 minus F at 3 divided by b minus A, so that's 3 minus -3, and this must be equal to the derivative of the function. At the given point C, right? In this case, we have to understand that our function is G, right? So essentially we can replace those F with G to essentially adjust the general equation of the mean value theorem to our example, and we're going to set our expression. Equal to G at point C and we want to identify those values of C. So what we have to do is value G of 3. We can use the given expression or we can use the graph, right? So if we look at G of 3, well, essentially, according to the given graph, we are going to get -3 as the value, right? So we can say that it is equal to -3. Now what about G of -3? Well, we can see that it corresponds to the value of 3. And this means that we're taking -3. Minus 3, divided by 3 + 3, right? So, basically we can say 3 + 3, and this must be equal to G C. Now, let's simplify. We're going to get -6 divided by 6, which is -1. And now if we go back to the expression of the derivative, we can say that G C. Which is equal to nec 2d divided by 3. Is equal to -1. And now we can solve for the possible values of C, right? If we multiply both sides by -3, we're going to get c 2d equals. Positive 3 and now there are two possible solutions. We're going to take square root of both sides. Let's recall that square root of C2 is positive or negative. See, so we're going to get C equals positive or negative square root of 3, and those are the possible c values. So let's Identify both of them. C1 is equal to square root of 3, and C2 is equal to square root of 3. Notice that we also need to check if they are within the given interval. Now C1 is greater than -3 but lower than 3, and C2 is also greater than -3 but lower than 3. So both of these values satisfy. The expression of the mean value theorem, and this means that we have two possible answers. C can be negative or positive square root of 3. Thank you for watching.

Key Concepts

Mean Value Theorem

Secant Line

Derivative

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