If you write out the equation of the tangent line of function f(x) at x=2, what are you writing?

Question

BlackTom AI · Accepted Answer

We begin by clarifying what the tangent line to f(x) at x = 2 represents. The tangent line is the linear approximation to f at the point (2, f(2)). Its slope is f'(2) and it passes through x = 2, so its standard point-slope form is y = f'(2)(x - 2) + f(2).

Option assessment, given there are no listed alternatives: the statement you provided says the tangent line is 'a function of x and no other variable.' In mathematical terms, the equation is indeed a function of the variable x, because the dependent quantity (the y-value on the tangent line) is determined by x through the expression y = f'(2)(x - 2) + f(2).

However, this description omits important context: the expression involves constants that come from the original function f, namely f(2) and f'(2). These are fixed numbers once f is specified, so the tangent line is a function of x with constant parameters.

To unpack potential misconceptions:
- If someone thinks the tangent line depends on variables other than x, that would be incorrect; there is no extra variable beyond x in the final line equation, aside from the constants derived from f.
- If someone insists the line cannot involve any constants, that would be inaccurate because any tangent line to a specific function at a specific point is determined by a slope and a point, both of which are constants for that particular x = 2.

In summary, the tangent line equation is a function of x, with constants f(2) and f'(2) embedded in it. The precise form is y = f'(2)(x - 2) + f(2). This emphasizes that while the variable is x, the line's particular coefficients come from the function f evaluated at 2 and its derivative at 2.

If you write out the equation of the tangent line of function f(x) at x=2, what are you writing?Single choice

Similar Questions

Which of the following is the equation of the line that is tangent to the curve 𝑦 = 5 𝑥 sin ⁡ ( 𝑥 ) at the point ( − 𝜋 , 0 ) ?

Which of the following is the equation of the line that is tangent to the curve 𝑦 = 4 𝑥 cos ⁡ ( 𝑥 ) at the point ( 3 𝜋 2 , 0 ) ?

Find an equation of the tangent line to the graph of at the point .

Suppose L is the tangent line of y = 3 tan ⁡ x − 2 csc ⁡ x at x = π 3 . Where does L intersect the x -axis?

Using your answer to the previous question, find an equation of the tangent line of 𝑓 ( 𝑥 ) = 𝑥 − 1 at 𝑥 = 5 . Write your answer in the form 𝑦 = 𝑚 𝑥 + 𝑏 .

Question text 2Marks What is the value of [math: x] where the tangent line to the curve [math: x2+y2+xy=3]x^2+y^2+xy=3 at the point [math: (1,1)] intersects the the [math: x]-axis (i.e. where [math: y=0])? Answer [math: x=] Answer 1[input].Notes Report question issue Question 27 Notes

Which ONE of the following thalamic nuclei is targeted by the trigeminothalamic tract?

Which ONE of the following is disinhibited in the direct pathway of the basal ganglia motor loop

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler