LEARNING FROM PAST
MISTAKES – MAT 271
Amanda Whitt
Asheville-Buncombe Technical Community College
Struggles
• ALGEBRA SKILLS!
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𝑥+2
𝑥+1
=2
• Application Problems (word problems)
• Understanding and applying definitions.
• Struggle to memorize “shortcuts”
• Confidence
Why Make or Use the Project?
• Review exams, labs, and graded assignments that have feedback.
• Helps students realize they need to continuously review material!
Project Overview
Take previously asked questions (or similar questions). Answer every question with the most
common, incorrect answer given by the students.
Project 1
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Topics:
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Limits
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Derivatives
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Product Rule, Quotient Rule, Chain
Rule
Logarithmic Differentiation
Implicit Differentiation
Related Rates
Project 2
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Topics:
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Analyzing graph of functions
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Increasing, decreasing, concavity,
minimums, maximums, etc…
l’Hospital’s Rule
Rolle's Theorem, Mean Value Theorem
Optimization
Riemann Sums
Definite, Indefinite Integrals
Fundamental Theorem of Calculus
Integration by substitution
The Project
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True or False:
If lim [𝑓 𝑥 𝑔 𝑥 ] exists, then the limit must be 𝑓 6 𝑔 6 .
𝑥→6
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True. Since we know the limit exists, then we use direct substitution to evaluate the
limit.
Use the definition of the derivative to find 𝑓′(4), where 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 4.
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𝑓 ′ 𝑥 = 2𝑥 − 3
𝑓′ 4 = 2 4 − 3 = 5
Problem 1
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True or False:
𝑓 𝑥 =
𝑥 2 −3𝑥+40
𝑥−8
and 𝑔 𝑥 = 𝑥 + 5 are equivalent functions.
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Explain your answer.
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Forget about domain
Get lost in notation
𝑥 2 −3𝑥+40
• lim
= lim 𝑥
𝑥−8
𝑥→8
𝑥→8
+5
Problem 2
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True or False:
If 𝑦 = 𝑒 2 , then 𝑦 ′ = 2𝑒.
Explain your answer.
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Students don’t think about the problem, just go through the motions.
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Still not sure about the number 𝑒
Problem 3
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True or False:
If 𝑓 𝑥 = 𝑥 6 − 𝑥 4 5 , then 𝑓 (31) = 0.
Explain your answer.
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Lack of algebra skills
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Lost in notation
Problem 4
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Find 𝑓′ in terms of 𝑔′. 𝑓 𝑥 = 𝑥 2 𝑔 𝑥
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Not sure about what is being asked
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Not comfortable with derivative rules
Problem 5
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Find the local and absolute extreme values of the function on the given interval.
𝑓 𝑥 = 𝑥 3 − 6𝑥 2 + 9𝑥 + 2, [2,4]
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Understand interval notation
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Forget to check the endpoints for extreme values
Problem 6
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Differentiate with respect to x. ℎ 𝑥 = sin x
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Students rely on Power Rule too much!
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Applying definitions.
x
Problem 7
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Find the general indefinite integral. (Use C for the constant of integration.)
4
3𝑥 + 𝑥 + 2
𝑑𝑥
𝑥 +1
2
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Forget about the constant “c”
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Do not recognize trigonometric integrals
Results
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Realize common mistakes and how to correct them
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Learn from past mistakes
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Build confidence
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Review material
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Helps the instructor enforce critical thinking questions for the students
Contact Me!
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[email protected]