AP Calculus AB / Contextual Applications of Differentiation / Question No. 33

33. Let $f$ be a differentiable function such that:

f (4) = 1, f^{'} (4) = 5

If the tangent line to the graph of $f$ at $x = 4$ is used to find an approximation to a zero of $f$ , that approximation is:

3.2

3.4

3.6

3.8

Answer is: option4

3.8

Solution:

The equation of the tangent line to $f (x)$ at $x = 4$ is given by the point-slope form:

y - f (4) = f^{'} (4) (x - 4)

Substituting the given values $f (4) = 1$ and $f^{'} (4) = 5$ :

y - 1 = 5 (x - 4)

y = 5 (x - 4) + 1

y = 5 x - 20 + 1

y = 5 x - 19

Finding the Zero

To approximate the zero of $f (x)$ , we find where the tangent line crosses the x-axis by setting $y = 0$ :

0 = 5 x - 19

Solving for $x$ :

5 x = 19

x = \frac{19}{5} = 3.8

Final Answer - 3.8

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