consider the following.\n\n$f(x)=3x^{2}-7$\n\nfind the following values of the function.\n\n$f(2 + delta…

consider the following.\n\n$f(x)=3x^{2}-7$\n\nfind the following values of the function.\n\n$f(2 + delta x)=$\n\n$f(2)=$\n\nfind the slope of the tangent line to the graph of the function at the point $(2,5)$.

consider the following.\n\n$f(x)=3x^{2}-7$\n\nfind the following values of the function.\n\n$f(2 + delta x)=$\n\n$f(2)=$\n\nfind the slope of the tangent line to the graph of the function at the point $(2,5)$.

Answer

Explanation:

Step1: Substitute ( x=2+\Delta x ) into ( f(x) )

( f(2+\Delta x) = 3(2+\Delta x)^2 - 7 )

Step2: Expand ( (2+\Delta x)^2 )

( (2+\Delta x)^2 = 4 + 4\Delta x + (\Delta x)^2 )

Step3: Simplify the expression

( 3(4 + 4\Delta x + (\Delta x)^2) - 7 = 3(\Delta x)^2 + 12\Delta x + 5 )

Step4: Substitute ( x=2 ) into ( f(x) )

( f(2) = 3(2)^2 - 7 )

Step5: Calculate ( f(2) )

( 3(4) - 7 = 5 )

Step6: Find derivative ( f'(x) )

( f'(x) = 6x )

Step7: Evaluate ( f'(2) )

( f'(2) = 6(2) = 12 )

Answer:

( f(2 + \Delta x) = 3(\Delta x)^2 + 12\Delta x + 5 ) ( f(2) = 5 ) The slope of the tangent line is 12