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

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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.

MAT136H1 S LEC0101 Online Practice 1

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

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