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

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
 .

BlackTom AI · Accepted Answer

We start by restating the problem: find the equation of the tangent line to f(x) = x − 1 at x = 5, written as y = mx + b, and choose from the given options.

Option 1: y = (1/4)x − 1/2. This line has slope m = 1/4 and intercept b = −1/2. Since the tangent line requires slope equal to f'(5) and passing through the point (5, f(5)) = (5, 4), this option would give y = (1/4)(5) − 1/2 = 5/4 − 1/2 = 5/4 − 2/4 = 3/4 at x = 5, which does not equal 4. Therefore, this cannot be the tangent line.

Option 2: y = (1/4)x + 3/4. Here again the slope is 1/4, which does not match the derivative f'(x) of f(x) = x − 1, since f'(x) = 1 for all x. Moreover, evaluating at x = 5 would give y = (1/4)(5) + 3/4 = 5/4 + 3/4 = 8/4 = 2, not 4. So this is incorrect as the tangent line.

Option 3: y = (1/2)x − 3/5. The slope is 1/2, not 1, so it cannot be the tangent line to f at x = 5. Checking the point: at x = 5, y would be (1/2)(5) − 3/5 = 5/2 − 3/5 = (25/10 − 6/10) = 19/10 = 1.9, which again does not match f(5) = 4. This option fails both slope and point criteria.

Option 4: y = (1/4)x − 3/5. Slope 1/4, not 1, so it contradicts the derivative. At x = 5, y = (1/4)(5) − 3/5 = 5/4 − 3/5 = (25/20 − 12/20) = 13/20 = 0.65, far from 4. This is not the tangent line.

Option 5: y = (1/2)x + 3/4. Slope 1/2; still not 1. The y-value at x = 5 would be y = (1/2)(5) + 3/4 = 5/2 + 3/4 = 10/4 + 3/4 = 13/4 = 3.25, which is not the function value at x = 5 (which is 4). So this cannot be the tangent line either.

Now, what is the correct tangent line? For f(x) = x − 1, the derivative f'(x) = 1 for all x, so the slope of the tangent at x = 5 is m = 1. The function value at x = 5 is f(5) = 5 − 1 = 4, so the tangent line passes through (5, 4) with slope 1. Using y = m(x − x0) + y0 with (x0, y0) = (5, 4) and m = 1 gives y = 1·(x − 5) + 4 = x − 5 + 4 = x − 1. Therefore the correct tangent line is y = x − 1.

Notice that none of the provided multiple-choice options matches y = x − 1, which is why the stated correct answer in the data (y = 1x − 1) aligns with the actual tangent line, but the answer options themselves do not include this exact form. The essential checks are the slope matching f'(5) = 1 and the line passing through (5, f(5)) = (5, 4).

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 .单项选择题

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