Test1 Review Material
Test Instructions
Below are the general instructions for all tests
- You will get a chance to retake this. Your highest score counts toward your final grade.
- Follow the guidance for each part and show all work for full credit.
- Non-graphical calculators are allowed.
- One page (two sides) of handwritten (or font size 8) notes are allowed.
- The equation sheet from the textbook is included as part of the test pack (it doesn’t count as note pages)
Practice Test 1 — Accumulation & Integrals
(Practice Only — No Solutions Included - Will Discuss on Wed)
Multiple Choice with Justification
Instructions:
Select the best answer and write one clear sentence justifying your choice.
Q1. Units and Interpretation
A sensor records a pollutant concentration change rate \(p(t)\) in mg/L per hour.
Which quantity has units of mg/L?
A. \(p(t)\)
B. \(\displaystyle \int_0^5 p(t)\,dt\)
C. \(\displaystyle \frac{1}{5}\int_0^5 p(t)\,dt\)
D. \(p(5) - p(0)\)
Q2. Net Change vs. Instantaneous Value
Let \(r(t)\) be a rate function.
Which statement is always true?
A. If \(r(t) = 0\) at some time, then the total accumulation is zero
B. A positive value of \(r(t)\) guarantees positive total accumulation
C. Total accumulation depends on both magnitude and duration of the rate
D. Negative values of \(r(t)\) cancel out and do not affect totals
Q3. Riemann Sums and Function Shape
Suppose \(f(x)\) is increasing and concave down on \([a,b]\).
Which statement is most accurate?
A. Left Riemann sums tend to underestimate the integral
B. Right Riemann sums tend to underestimate the integral
C. Trapezoidal rule will always overestimate the integral
D. All Riemann sums give the same result
E. None of the above
Q4. What Improves with More Subintervals?
When increasing the number of subintervals in a Riemann sum approximation, what improves directly?
A. The algebra becomes simpler
B. The approximation more closely reflects local behavior of the function
C. The integral becomes exact
D. The function becomes linear on each subinterval
Q5. Midpoint Rule Reasoning
Why can the midpoint rule outperform left- or right-hand sums for many functions?
A. It uses more subintervals
B. It assumes the function is constant
C. It samples where over- and under-estimates may balance
D. It eliminates the need for limits
Q6. Linearity and Constants
Which expression is equivalent to
\[
\int_0^2 (x^2 + 5)\,dx ?
\]
A. \(\int_0^2 x^2\,dx + 5\)
B. \(\int_0^2 x^2\,dx + \int_0^2 5\,dx\)
C. \(\int_0^2 (x^2)\,dx \cdot 5\)
D. \(5\int_0^2 x^2\,dx\)
Q7. Zero Accumulation
Which situation guarantees
\[
\int_a^b f(x)\,dx = 0 ?
\]
A. \(f(x) = 0\) at one point in \([a,b]\)
B. \(f(x)\) is sometimes positive and sometimes negative
C. The signed areas above and below the axis cancel exactly
D. \(f(x)\) is decreasing on \([a,b]\)
Q8. Average Value of a Function
Which expression represents the average value of \(f(x)\) on \([a,b]\)?
A. \(f(b) - f(a)\)
B. \(\displaystyle \int_a^b f(x)\,dx\)
C. \(\displaystyle \frac{1}{b-a}\int_a^b f(x)\,dx\)
D. \(\displaystyle \frac{f(a)+f(b)}{2}\)
Problem 1: Comparing Approximation Methods \([15 pts]\)
A rate function is given by
\[
f(x) = 3 - x
\]
on the interval \([0,4]\).
(a) Left-Hand Riemann Sum
Compute the left-hand Riemann sum for \(f(x)\) on \([0,4]\) using
\[
n = 4
\]
equal subintervals.
7.11.12 (b) Right-Hand Riemann Sum
Compute the right-hand Riemann sum for the same function and interval using
\[
n = 4
\]
equal subintervals.