Question

Standardized Test Questions You may use a graphing calculator to solve the following problems. True or False If \(f\) is continuous on an open interval \(I\) containing \(a,\) then \(F\) defined by \(F(x)=\int_{a}^{x} f(t) d t\) is continuous on \(I .\) Justify your answer.

Short answer

The statement is True.

Step-by-step solution

Recognising the Concept

This problem is about the continuity of integrals. It involves linking the continuity of a function and the continuity of its integral. The function \(f(t)\) is given as continuous on an open interval \(I\), and a function \(F(x)\) is defined by using integral of \(f(t)\) from \(a\) to \(x\). The problem is to analyse if \(F(x)\) is continuous on \(I\).

Applying the Fundamental Theorem of Calculus

By applying the Fundamental Theorem of Calculus Part 1, if \(f\) is continuous on an interval \(I\), then the function \(F(x)\) defined by \(F(x)=\int_{a}^{x} f(t) dt\) for \(x\) in \(I\) is also continuous on \(I\). A direct result from this theorem is that if \(f\) is continuous on the open interval \(I\) containing \(a\), then the function \(F(x)\) defined as \(F(x)=\int_{a}^{x} f(t) dt\) is continuous on \(I\).

Justification

Therefore, the statement 'If \(f\) is continuous on an open interval \(I\) containing \(a,\) then \(F\) defined by \(F(x)=\int_{a}^{x} f(t) dt\) is continuous on \(I\)' is True. This is because any definite integral of a continuous function will produce a continuous function on the given interval.

Key concepts

Fundamental Theorem of Calculus

Understanding the link between differentiation and integration is crucial when studying calculus. This connection is encapsulated in the Fundamental Theorem of Calculus (FTC), a pivotal idea that bridges the two essential operations. The FTC articulates that if you have a continuous function over an interval, you can obtain its antiderivative, and use it to evaluate definite integrals.

The theorem has two main parts: The First Part states that if function f is continuous over an interval \[I\], then its integral, \[F(x)\], is a differentiable function over \[I\], and the derivative of \[F(x)\] is \[f(x)\]. The Second Part allows us to evaluate the integral of \[f(x)\] on the interval from \[a\] to \[b\] by subtracting the values of its antiderivative at \[b\] and \[a\]. Simply put, it provides a convenient method to calculate the area under the curve represented by f.

In essence, this theorem can be seen as a grand reveal: every continuous 'accumulation' process (integration) can be associated with a precise 'rate of change' process (differentiation), and vice versa.

Continuous Function

A continuous function is one that has no interruptions, jumps, or breaks within its domain. Intuitively, it means you can draw the graph of the function without lifting your pen off the paper. Formally, a function \[f\] is continuous at a point \[x=a\] if three conditions are met:

• The function \[f(a)\] is defined,
• The limit of \[f(x)\] as \[x\] approaches \[a\] exists,
• The limit of \[f(x)\] as \[x\] approaches \[a\] is equal to \[f(a)\].

When a function is continuous over an interval, it's devoid of any instances where these conditions are not fulfilled. This smoothness is what ensures that when we integrate a continuous function, the resulting function, namely the integral, is also continuous. Continuity is a favorable property when working with integrals because it guarantees no unexpected behavior within the range of integration.

Graphing Calculator

A graphing calculator is a student's BFF when it comes to visualizing mathematical concepts, especially in standardized testing environments. This powerful tool not only performs basic arithmetic but also plots graphs, solves equations, and simulates calculations involving calculus operations. By inputting a function into a graphing calculator, students can readily see its graphic representation. Moreover, for continuous functions, one can use the calculator to find areas under curves (numerical integration) or even assist with understanding the behavior of functions thanks to the zoom and trace features.

For students tackling integrals or the Fundamental Theorem of Calculus, a graphing calculator can be a helpful ally. It can visually affirm the continuity of \[F(x)\] resulting from the integration of \[f(x)\], and thus, corroborate their solutions. In addition, during standardized tests, when time is of the essence, the calculator's functions can expedite problem-solving and provide graphical confirmation of continuity or other function properties.

Standardized Test Questions

Standardized test questions are designed to assess students' comprehension and proficiency across a range of topics under equal conditions. In the context of calculus, these questions often cover concepts like the Fundamental Theorem of Calculus and continuous functions—core ideas that constitute the building blocks of higher mathematics.

When such topics appear in standardized test questions, students must not only understand the theoretical foundation but also how to apply these concepts practically. In many cases, these tests also allow the use of a graphing calculator, enabling students to solve complex problems more efficiently and accurately. Hence, familiarizing oneself with the application of graphing calculators to these concepts proves to be invaluable preparation.

Students should practice a variety of problems, including those that ask for graphical understanding, algebraic manipulation, and logical reasoning, to ensure a strong command over these concepts and succeed in standardized assessments.