First find the domain of the given function f and then find...

First find the domain of the given function $f$ and then find where it is increasing and decreasing, and also where it is concave upward and downward. Identify all extreme values and points of inflection. Then sketch the graph of $y=f(x)$
$$f(x)=\int_{1}^{x} 2^{-t^{2}} d t$$

Key Concepts

Graph Sketching Techniques

Graph sketching involves synthesizing information from the domain, first and second derivative tests, and other function properties to produce a qualitative representation of the function's behavior. This includes noting where the function increases or decreases, its concavity, extreme values, and inflection points to construct an accurate sketch of the graph.

Concavity and Points of Inflection

Determining concavity involves analyzing the second derivative of the function. If the second derivative is positive over an interval, the function is concave upward (convex), and if it is negative, the function is concave downward (concave). Points of inflection occur where the concavity changes, signaling a change in the curvature of the graph.

Extreme Value Analysis

This concept deals with identifying local minima and maxima using the first derivative test and, if needed, the second derivative test. By finding where the derivative is zero or undefined (critical points) and analyzing the sign changes around these points, one can classify the nature of these extreme values.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative is equal to the integrand evaluated at that upper limit. This property is essential for analyzing the behavior of integral-defined functions by allowing us to determine the first and second derivatives.

Domain of Integral-Defined Functions

This concept involves determining the set of x-values for which a function defined by an integral is valid. For functions like this, the domain is typically considered where the integrand is well-defined and the limits of integration make sense, often resulting in the function being defined for all real numbers if the integrand is continuous everywhere.

Analysis of Increasing and Decreasing Functions

This concept involves determining where a function is increasing or decreasing by studying the sign of its first derivative. A positive first derivative indicates that the function is increasing over that interval, while a negative value indicates it is decreasing. This is crucial for identifying potential extreme values.

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