Question

Calculus 1
Is my shown work correct?

Find an equation of the line that is tangent to the graph of f and parallel to the given line.

I'll post the equations and my work so far done in the comments

Answer 1

Find an equation of the line that is tangent to the graph of f and parallel to the given line.
\[f(x) =x^{3}+2 \]
and
\[3x - y + 1 = 0\]
This is my solution. Is it correct?
\[f^{'}= 3x^2\]
given line y =3x + 1 m=3

To find where the 2 slopes are equal set f’ prime = m

\[3x^{2}= 3\]
both sides divided by 3
\[x^{2}=1\]
square root of both sides
\[x^{2}=\pm1\]

Since 1 is a value for x and y = f(x) To determine y, 1 can be plugged into f(x) =
\[(1)^{3}+2 = 5\]
So we have the coordinates (1,5) as well as m = 3. I used
\[y_{2}-y _{1}= m(x _{2}-x _{1})\]
y - 5 = 3(x -1)

y - 5 = 3x -3

y = 3x +4

Answer 2

y = 3x+4 is one of the solutions

there is another tangent line

Answer 3

your work looks good so far though