Q. 42 - Calculus

Question

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{\text{'}}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=\arctan x,c=0$

Step-by-Step Solution

Answer

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Step 1: Use difference quotient

Approximate \( f'(c) \) using \( \frac{f(c+h) - f(c)}{h} \) for decreasing values of \( h \).

Step 2: Known derivative

For \( f(x) = \arctan(x) \), the exact derivative is \( f'(x) = \frac{1}{1+x^2} \). The secant lines approach the tangent line as \( h \to 0 \).