ww5: problem 14 (1 point) results for this submission entered answer preview result 0.0286 0.0286 incorrect…

ww5: problem 14 (1 point) results for this submission entered answer preview result 0.0286 0.0286 incorrect the answer above is not correct. use the figures below to evaluate the indicated derivative or state that it does not exist. if the derivative does not exist, enter dne in the answer blank. the graph to the left (in black) gives f(x), while the graph to the right gives g(x) (which is constant for values of x greater than 60). d/dx f(g(x))|x = 30 (if the derivative does not exist, enter dne.) preview my answers submit answers your score was recorded. you have attempted this problem 4 times. you received a score of 0% for this attempt. your overall recorded score is 0%. you have unlimited attempts remaining.

ww5: problem 14 (1 point) results for this submission entered answer preview result 0.0286 0.0286 incorrect the answer above is not correct. use the figures below to evaluate the indicated derivative or state that it does not exist. if the derivative does not exist, enter dne in the answer blank. the graph to the left (in black) gives f(x), while the graph to the right gives g(x) (which is constant for values of x greater than 60). d/dx f(g(x))|x = 30 (if the derivative does not exist, enter dne.) preview my answers submit answers your score was recorded. you have attempted this problem 4 times. you received a score of 0% for this attempt. your overall recorded score is 0%. you have unlimited attempts remaining.

Answer

Explanation:

Step1: Recall chain - rule

The chain - rule states that if (y = f(g(x))), then (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). We need to find (f^\prime(g(30))\cdot g^\prime(30)).

Step2: Analyze (g(x)) graph

From the graph of (g(x)), for (x = 30), the function (g(x)) is a straight - line segment. The slope of a non - vertical line (y=mx + b) is given by (m=\frac{\Delta y}{\Delta x}). The graph of (g(x)) is a straight line for (x) values we are interested in. The slope (g^\prime(x)) is constant for the non - horizontal part of the graph. For (g(x)), if we consider two points on the line of (g(x)) (say ((0,20)) and ((60,60))), then (g^\prime(x)=\frac{60 - 20}{60-0}=\frac{2}{3}).

Step3: Analyze (f(x)) graph

First, find (g(30)). Since (g(x)) is a linear function (g(x)=20+\frac{2}{3}x) (using the point - slope form (y - y_1=m(x - x_1)) with (m=\frac{2}{3}) and ((x_1,y_1)=(0,20))), then (g(30)=20+\frac{2}{3}\times30=20 + 20=40). Then, find the slope of (f(x)) at (x = 40). The graph of (f(x)) is a straight line for (x\geq40). The two points on the right - hand side line of (f(x)) are ((40,0)) and ((60,20)). The slope of (f(x)) for (x\geq40) is (f^\prime(x)=\frac{20 - 0}{60 - 40}=1).

Step4: Apply chain - rule

By the chain - rule (\left.\frac{d}{dx}f(g(x))\right|{x = 30}=f^\prime(g(30))\cdot g^\prime(30)). We found that (g(30)=40), (f^\prime(g(30))=f^\prime(40)=1) and (g^\prime(30)=\frac{2}{3}). So (\left.\frac{d}{dx}f(g(x))\right|{x = 30}=1\times\frac{2}{3}=\frac{2}{3}).

Answer:

(\frac{2}{3})