Learning Journal:
Write 2 true/false questions that illustrate a variety of ideas from this week’s topics that
you might put on this exam if you were teaching the class.

Give a key

Explain the answers

Explain why you chose these particular questions and what the questions will assess
Question 1: True/False
The definite integral of a function represents the area under the curve of the function’s derivative.
Key: False
Explanation: The statement is false. The definite integral of a function represents the area under
the curve of the function itself, not its derivative. The definite integral is a way to calculate the
accumulated value of a function over a specified interval. It measures the net area between the
curve and the x-axis within the given interval. On the other hand, the indefinite integral gives us
a family of functions whose derivatives are equal to the original function. The Fundamental
Theorem of Calculus establishes a connection between the definite integral and the antiderivative,
stating that the definite integral of a function can be evaluated using the antiderivative at the
interval endpoints.
Assessment: This question assesses the student’s understanding of the difference between
definite and indefinite integrals, as well as their grasp of the Fundamental Theorem of Calculus. It
requires students to correctly identify the relationship between integrals and derivatives,
demonstrating their ability to apply these concepts to different scenarios.
Question 2: True/False
When approximating the area under a curve using rectangles, increasing the number of rectangles
will always result in a more accurate approximation.
Key: True
Explanation: The statement is true. The process of approximating the area under a curve using
rectangles is known as Riemann Sum approximation. As the number of rectangles (or subintervals)
used in the approximation increases, the width of each rectangle becomes narrower, resulting in
a better alignment with the curve. This leads to a more accurate estimation of the actual area
under the curve. In the limit, as the number of rectangles approaches infinity, the Riemann sum
becomes the definite integral, providing the exact value of the area.
Assessment: This question evaluates the student’s understanding of the relationship between the
number of rectangles and the accuracy of Riemann Sum approximations. It tests whether the
student comprehends the concept of the limit as the number of rectangles increases and its
connection to the definite integral. Additionally, the question addresses the idea of convergence,
a key concept in calculus.
These particular questions were chosen to cover a range of ideas from this week’s topics. The first
question examines the understanding of the Fundamental Theorem of Calculus and the distinction
between definite and indefinite integrals. It tests the student’s ability to differentiate between
functions and their derivatives in the context of integration. The second question focuses on the
practical application of approximating areas under curves using Riemann Sums. It gauges the
student’s grasp of the relationship between the number of rectangles and the accuracy of the
approximation, while also assessing their understanding of limits and convergence. Together,
these questions provide a well-rounded assessment of the key concepts covered in this week’s
unit on indefinite integrals, the definite integral, integration formulas, and the Fundamental
Theorem of Calculus.
Reference
Herman,
E.
&
Strang,
G.
(2020). Calculus
volume
1.
OpenStacks.
University. https://openstax.org/books/calculus-volume-1/pages/2-introduction
Rice