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, 3x - y - 4 = 0` Find an equation of the line that is tangent to the graph of f and parallel to the given line.
Since the tangent line to function f(x) = x ^3 + 2 has to be parallel to the line 3x - y - 4 = 0, both needs to have the same slope.
Equation of the straight line can be rewritten in slope-intercept form to : y = 3x - 4. Hence the desired slope of our line is 3.
And derivative of f(x) will provide the slope of the line tangent to f(x)
d/dx(f(x)) = 3x
setting it equal to 3
3x = 3 or x = 1 provides one of the conditions
Corresponding y coordinate can be found from the f(x) = x^3 + 2 equation, y = (1)^3 + 2 = 3
The slope of the desired line is equal to 3, and we now have a point on the desired line (1, 3), as well the point of tangency
Using the point-slope, equation to desired line can be obtained as follows. Given slope m = 3 and a point on the desired line,(x0, y0), the equation of the line is given by
(y - y0) = m(x - x0)
y - 3 = 3(x - 1 )
y = 3x
The slope of the line 3x-y-4=0 and the lines parallel to it will be same.
Therefore the slope(m) of the line is the coefficient of the x = 3
The slope of the tangent line to the graph f(x) is the derivative of the f(x)
Since the tangent line to the graph is parallel to the given line
The y coordinate can be found by substituting the value of x in the function.
for x=1 y=1^3+2=3
for x=-1 y=(-1)^3+2=1
The equation of the tangent line can be written using the point slope form of the equation.