Estimating Derivatives - Calculus 1

1. Let $f(x)$ be a function with $f''(x)<0$ for all $x$. An estimate for $f'(2)$ is calculated using the formula $E = \frac{f(2.1) - f(2)}{0.1}$. Which of the following statements about the estimate $E$ must be true?
   Answer: $E < f'(2)$ The condition $f''(x)<0$ means that the function $f(x)$ is concave down. For a concave down function, the graph lies below its tangent lines. Geometrically, this means the slope of the function is always decreasing. The estimate $E$ is the slope of the secant line connecting the points $(2, f(2))$ and $(2.1, f(2.1))$. Because the function's slope is decreasing over the interval $[2, 2.1]$, the slope of the tangent line at the start of the interval, $f'(2)$, will be greater than the slope of the secant line over the interval. Therefore, $E < f'(2)$. The estimate is an underestimate.
2. Consider the following three expressions used to estimate $f'(2)$ for a smooth function $f(x)$: I. $E_1 = \frac{f(2.1) - f(2)}{0.1}$ II. $E_2 = \frac{f(2.01) - f(1.99)}{0.02}$ III. $E_3 = \frac{f(3) - f(1)}{2}$ Which of the following correctly orders these estimates from typically least accurate to typically most accurate?
   Answer: III, I, II We need to compare the accuracy of the three estimates. Accuracy is determined by two factors: the type of difference quotient and the size of the interval ($h$). $E_3$ is a symmetric difference quotient, but its interval is very wide ($[1, 3]$, so $h=1$). This will typically make it the least accurate estimate for the instantaneous rate of change at $x=2$. $E_1$ is a forward difference quotient with a small interval ($h=0.1$). This is better than $E_3$. $E_2$ is a symmetric difference quotient with a very small interval ($h=0.01$). Symmetric difference quotients are generally more accurate than one-sided quotients, and smaller values of $h$ are generally more accurate than larger values. Thus, $E_2$ is the most accurate estimate. The order from least accurate to most accurate is therefore $E_3$, then $E_1$, then $E_2$.
3. Let $f(x)$ be a differentiable function for which the second derivative $f''(x)$ exists. Which of the following expressions generally provides the most accurate estimate of $f'(a)$?
   Answer: $\frac{f(a+0.01) - f(a-0.01)}{0.02}$ The accuracy of a finite difference approximation to a derivative depends on both the step size $h$ and the type of difference formula used. The expressions in A, B, and C are one-sided (forward or backward) difference quotients. The expression in D is a symmetric (or central) difference quotient. For a function that is sufficiently smooth, the error in a one-sided difference is proportional to the step size $h$ (i.e., error is $O(h)$). The error in a symmetric difference is proportional to the square of the step size (i.e., error is $O(h^2)$). Because the error in the symmetric difference formula decreases much faster as $h \to 0$, it is generally much more accurate than a one-sided difference, even if the step size in the one-sided formula is smaller. Therefore, the symmetric difference in choice D provides the best estimate.

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