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

33. Let \( f \) be a differentiable function such that:

\[ f(4) = 1, \quad f'(4) = 5 \]

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 \( f'(4) = 5 \):

\[ y - 1 = 5(x - 4) \] \[ y = 5(x - 4) + 1 \] \[ y = 5x - 20 + 1 \] \[ y = 5x - 19 \] Finding the Zero

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

\[ 0 = 5x - 19 \]

Solving for \( x \):

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

Final Answer - 3.8

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