3.1 Average vs Instantaneous Rate of Change (#limitscontinuity, #differentiation) Consider the piecewise function \begin{cases} 2x + 5 & x \le -2 \\ sin(\pi x) + 1 & -2 < x < 2 \\ 2 & x = 2 \\ 1 - \frac{1}{x} & x > 2 \end{cases} (a) Graph $f$. (b) Find the average rate of change of $f$ from $x = -3$ to $x = 1$ and $x = 0$ to $x = 1$. Add the corresponding secant lines to your graph. (c) Find the instantaneous rate of change of $f$ at $x = 1$ and add the corresponding tangent line to your graph. (d) Where is $f$ discontinuous? (e) Where is $f$ not differentiable?
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- For \( x \le -2 \), use \( f(x) = 2x + 5 \). - For \( -2 < x < 2 \), use \( f(x) = \sin(\pi x) + 1 \). - For \( x = 2 \), \( f(x) = 2 \). - For \( x > 2 \), use \( f(x) = 1 - \frac{1}{x} \). Show more…
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