Let F(x)=∫a^u(x) f(t) d t for the specified a, u, and f ....

Let $F(x)=\int_{a}^{u(x)} f(t) d t$ for the specified $a, u,$ and $f .$ Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of $F .$ b. Calculate $F^{\prime}(x)$ and determine its zeros. For what points in its domain is $F$ increasing? Decreasing? c. Calculate $F^{\prime \prime}(x)$ and determine its zero. Identify the local extrema and the points of inflection of $F .$ d. Using the information from parts $(a)-(c),$ draw a rough hand-sketch of $y=F(x)$ over its domain. Then graph $F(x)$ on your CAS to support your sketch. $$ a=0, \quad u(x)=x^{2}, \quad f(x)=\sqrt{1-x^{2}} $$

Let $F(x)=\int_{a}^{u(x)} f(t) d t$ for the specified $a, u,$ and
$f .$ Use a CAS to perform the following steps and answer the questions
posed.
a. Find the domain of $F .$
b. Calculate $F^{\prime}(x)$ and determine its zeros. For what points in its
domain is $F$ increasing? Decreasing?
c. Calculate $F^{\prime \prime}(x)$ and determine its zero. Identify the local
extrema and the points of inflection of $F .$
d. Using the information from parts $(a)-(c),$ draw a rough hand-sketch of $y=F(x)$ over its domain. Then graph $F(x)$ on your CAS to support your sketch.
$$
a=0, \quad u(x)=x^{2}, \quad f(x)=\sqrt{1-x^{2}}
$$

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, then its derivative is given by the integrand evaluated at that upper limit, multiplied by the derivative of the upper limit (by the chain rule). This concept is crucial when differentiating functions of the form F(x) = ?[a, u(x)] f(t) dt.

Chain Rule in Differentiation

The chain rule is a basic rule in calculus used to differentiate composite functions. When a function is defined as an integral with a variable upper limit, the derivative involves applying the chain rule to account for the dependence of the upper limit u(x) on x, effectively multiplying the derivative of the upper limit by the integrand evaluated at u(x).

Domain of Functions Defined by Integrals

Determining the domain of a function defined by an integral involves considering both the limits of integration and the domain of the integrand. Since the integral is taken from a fixed lower limit to a variable upper limit, the domain of F(x) is constrained by the values of x for which the upper limit u(x) produces a valid input for the integrand f(t).

First Derivative and Monotonicity Analysis

Analyzing the first derivative of a function is fundamental for understanding its monotonic behavior. By finding where the first derivative is positive, negative, or zero, one can determine the intervals where the function is increasing or decreasing, as well as identify potential local extrema. This involves solving F'(x) = 0 and checking the sign of F'(x) across different intervals.

Second Derivative and Concavity Analysis

The second derivative provides information about the concavity of a function and the nature of its curvature. By determining where F''(x) is positive or negative, one can identify regions where the function is concave upward or downward. Additionally, setting F''(x) = 0 helps locate points of inflection, which are points where the concavity changes.

Graph Sketching Based on Derivative Information

Graph sketching synthesizes information from the domain, first and second derivative analyses to create a rough visualization of a function's behavior. By identifying key features such as increasing/decreasing intervals, local extrema, concavity, and inflection points, one can draw a hand-sketched graph that reflects the overall shape and important characteristics of the function.

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