If f(2)=3 and f'(2)=5, write an equation for the tangent line to the graph of y=f(x) at the point where x=2.

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to write the equation of the tangent line, hence you need at least two points to write this equation suct that:

`y - y_1 = (y_2 - y_1)/(x_2 - x_1) * (x - x_1)`

Notice that the problem provides the derivative of the function at the point x = 2. You need to link this information to the fact that the derivative of a function at a point is the tangent line to the graph of function at the point, such that:

f'(x) = m (m denotes the slope of the line or the tangent of angle made by line to x axis)

You need to remember what tangent function is:

tan `alpha`  = opp side/adj side

You need to find the length of orthogonal projection to y axis and x axis such that:

`l_1 ` = opp side`= y_2 - y_1`

`` `l_2`  = adj side `= x_2 - x_1`

Hence, `m = (y_2 - y_1)/(x_2 - x_1)`

Notice that the point (2,3) is on the graph of f(x) and on the tangent line such that:

`m = (y_2 - y_1)/(x_2 - x_1) = 5`

You may write the new form of equation of tangent line:

`y = 5(x-2) + 3 =gt y = 5x - 7`

Hence, the equation of tangent line is y = 5x - 7.

nathanshields's profile pic

nathanshields | High School Teacher | (Level 1) Associate Educator

Posted on

Remember point-slope form for a line:

`y-y_1=m(x-x_1)`

We can fill in all we need from the given information.

y - 3 = 5(x - 2)

This answer is fine, but if you need to you can put it into slope-intercept form:

y-3 = 5x - 10

y = 5x - 7

 

 

 

 

 

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