GU Sketch the graph of f(x)=x^5 / 2 on [0,6] (a) Use the...

GU Sketch the graph of $f(x)=x^{5 / 2}$ on [0,6] (a) Use the sketch to justify the inequalities for $h>0$ : $$ \frac{f(4)-f(4-h)}{h} \leq f^{\prime}(4) \leq \frac{f(4+h)-f(4)}{h} $$ (b) Use (a) to compute $f^{\prime}(4)$ to four decimal places. (c) Use a graphing utility to. plot $y=f(x)$ and the tangent line at $x=4$, utilizing your estimate for $f^{\prime}(4)$.

GU Sketch the graph of $f(x)=x^{5 / 2}$ on [0,6] (a) Use the sketch to justify the inequalities for $h>0$ :
$$
\frac{f(4)-f(4-h)}{h} \leq f^{\prime}(4) \leq \frac{f(4+h)-f(4)}{h}
$$
(b) Use (a) to compute $f^{\prime}(4)$ to four decimal places.
(c) Use a graphing utility to. plot $y=f(x)$ and the tangent line at $x=4$, utilizing your estimate for $f^{\prime}(4)$.

Key Concepts

Power Functions with Fractional Exponents

Power functions with fractional exponents, like the one analyzed in the problem, exhibit unique behaviors such as non-integer growth rates and specific curvature properties. Understanding these functions helps in analyzing their derivatives and graph shapes, which is crucial for justifying the derivative approximations and constructing the tangent line.

Tangent Line Approximation

The tangent line at a point on a function serves as a linear approximation of the function near that point. Its slope, which is the derivative at that point, provides the best linear estimate of the function, and graphing tools can be used to visualize this approximation in relation to the overall curve.

Graphical Analysis

Graphical analysis involves sketching or plotting the function to visually assess its behavior, such as increasing/decreasing intervals and slopes of tangents. This approach helps justify analytical bounds and inequalities by providing a visual representation of how the secant lines relate to the tangent line at a specific point.

Difference Quotient

The difference quotient is used to approximate the derivative by calculating the average rate of change over a finite interval. Using values from either side of a point, left-hand and right-hand difference quotients provide estimates that can be compared to establish bounds for the actual derivative, especially useful in justifying inequalities for h > 0.

Derivative

The derivative of a function at a point is the measure of its instantaneous rate of change and is defined as the limit of the difference quotient as the interval approaches zero. It is fundamental in calculus as it connects the algebraic expression of a function to the geometric concept of the slope of the tangent line at that point.

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