if ( f(x)=3x^{2}-7x + 4 ), find ( f(-5) ).\nuse this to find the equation of the tangent line to the…

if ( f(x)=3x^{2}-7x + 4 ), find ( f(-5) ).\nuse this to find the equation of the tangent line to the parabola ( y = 3x^{2}-7x + 4 ) at the point ( (-5,114) ). the equation of this tangent line can be written in the form ( y=mx + b )\nwhere ( m ) is:\nand where ( b ) is:\nquestion help: video message instructor\nsubmit question jump to answer

if ( f(x)=3x^{2}-7x + 4 ), find ( f(-5) ).\nuse this to find the equation of the tangent line to the parabola ( y = 3x^{2}-7x + 4 ) at the point ( (-5,114) ). the equation of this tangent line can be written in the form ( y=mx + b )\nwhere ( m ) is:\nand where ( b ) is:\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Find the derivative of (f(x))

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (f(x)=3x^{2}-7x + 4), we have (f^\prime(x)=(3x^{2})^\prime-(7x)^\prime+(4)^\prime). (f^\prime(x)=3\times2x-7\times1+0 = 6x-7).

Step2: Calculate (f^\prime(-5))

Substitute (x = - 5) into (f^\prime(x)). (f^\prime(-5)=6\times(-5)-7=-30 - 7=-37). So (m=-37).

Step3: Find the equation of the tangent line (y = mx + b)

We know the point ((x_0,y_0)=(-5,114)) and (m=-37). Substitute into (y=mx + b) (i.e., (y_0=mx_0 + b)). (114=-37\times(-5)+b). (114 = 185 + b). Solve for (b): (b=114 - 185=-71).

Answer:

(f^\prime(-5)=-37), (m=-37), (b=-71)