Find an equation of the tangent line to the graph of the function f defined by the given equation at the point .

Question

Find an equation of the tangent line to the graph of the function f defined by the given equation at the point  . ​  ​

Quizplus · Accepted Answer

We need to find the derivative of f at the given point to get the slope of the tangent line. 
Using the power rule, we get:
f'(x) = 3x^2 - 4x + 1
f'(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) = 3(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)^2 - 4(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) + 1

Simplifying this expression, we get:
f'(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) = 3(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)^2 - 44ebc3a0_8643_5c06_a16e_cdbd0a88167b_TB8255_11 + 1

Now we can use the point-slope form of the equation of a line to find the tangent line. 
y - f(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) = f'(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)(x - 11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)

Plugging in the values we found, we get:
y - 11eb651f_cb63_5c0a_b0a1_a5c15b7fe69d_TB8255_11 = (3(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)^2 - 44ebc3a0_8643_5c06_a16e_cdbd0a88167b_TB8255_11 + 1)(x - 11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)

Simplifying this expression, we get:
y = 3x - 3  (Choice C)

Find an Equation of the Tangent Line to the Graph