Use the figure below to estimate the indicated derivatives,...

Use the figure below to estimate the indicated derivatives, or state that they do not exist. If a derivative does not exist, enter dne in the answer blank. The graph of f(x) is black and has a sharp corner at x = 2. The graph of g(x) is blue. Let h(x) = f(x) · g(x). Find A. h'(1) = B. h'(2) = dne C. h'(3) =

Transcript

00:01 Consider the given graph of f of x and g of x below, and suppose h of x is equal to the product of f of x and g of x.

00:11 Now, in here we want to find the values of h prime of 1, h prime of 2, and h prime of 3.

00:19 Now, to do that, we first want to find the derivative of h of x in general.

00:27 So by product rule, h prime of x, this is just f of x times g prime of x plus g of x times f prime of x.

00:39 Now we note that the values of g prime of x and f prime of x are the slopes of the tangent lines to g of x and f of x at x.

00:54 So if we want to find h prime of 1, which is just f of 1 times g prime of 1, plus g of 1 times f prime of 1, we will first find the values of f of 1, g prime of 1, g of 1, g of 1, g of 1 and f prime of 1, and then later we will combine.

01:15 Now, f of 1 is just 1 .5 based on the graph.

01:22 So we will have here 1 .5 .5.

01:24 5 times you have g prime of 1, which is the slope of the tangent line.

01:31 Now, since this is already a line, then the tangent line and this line will have the same slope wherever your x is.

01:39 So in this case, it will be a rise over run.

01:44 That's one unit downward, so we have negative 1 over.

01:49 We have 4 units to the right.

01:52 That will be positive 4.

01:55 Plus we have g of 1 that is the value of g when x is 1 and based on this graph this is 2 over 3 times f prime of 1 which is the slope of the tangent line to f of x at x equals 1 so we can use the point 0 0 and this point here to find the slope so that's going to be our rise of three units and our run of two units towards the right so that's a positive two.

02:41 Now combining we have 1 .5 times negative 1 over 4 that's negative 3 over 8 plus we have 2 thirds times 3 halves that's 1.

02:55 Combining further this gives us a value equal to 5 over 8 so this is our h prime of 1.

03:04 Okay, next we want to find h prime of 2, so the same process we do f of 2 times g prime of 2 plus g of 2 times f prime of 2.

03:19 Now f of 2, this is the value of f when x is equal to 2 and based on the graph that's equal to 3 times g prime of 2 that will be the same negative 1 over 4 since we are looking at the same line.

03:37 So that's 1 unit downwards and 4 units to the right.

03:41 That's why it's negative 1 over 4.

03:43 Plus we have g of 2.

03:46 The value of g when x is 2, that will be 0 .5 or 1⁄2.

03:54 And then lastly we have f prime of 2...

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