Assume f and g are differentiable on their domains with h(x)=(fog)(r). Suppose the equation of the line tangent to the graph of g at the point (4,7) isy=3x–5 and the equation of the line tangent to the graph of f at (7,9) is y=−2x+23. (a) Calculate h(4) and h′(4). (b) Determine an equation of the line tangent to the graph of h at the point on the graph where x=4.

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Melinda McCombs · Accepted Answer

Step 1:Define the functionsNSk(a)Since h(4)=f(g(4)) and h′(4)=f′(g(4)).g′(4) , therefore, to determine the value of h(4) and h′(4) : we need to know the values of g(4),f(g(4)),g′(4) and f′(g(4)) .Now, (4,7) point lies on the graph of g, therefore, g(4)=7.Similarly, (7,9) lies on graph f , therefore, f(7)=9.Now, g(4)=7 , therefore, f(g(4))=9.We know that if a line is tangent to a graph at a point (x,y), then differentiation of the function will have the same value to the slope of the line.Now, slope of the line : y=3x−5 is 3.As the the line y=3x−5 is tangent to the graph g on (4,7), therefore, g′(4)=3.Similarly,we have, slope of the line y=−2x+23 is -2 .Therefore, f′(7)=−2.Therefore, f′(g(4))=−2.So,h(4)=f(g(4))=9.And,h′(4)=f′(g(4)).g′(4)=−2.3=−6.

Neil Dismukes · Accepted Answer

Step 2: Second Part(b)Using the formula of tangent , we have :y−h(4)=h′(4).(x−4)    [atx=4]⇒y−9=(−6).(x−4)⇒y=−6x+33.Therefore, equation of the tangent line is : y=−6x+33.

Assume f and g are differentiable on their domains with h

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