How does the fundamental theorem evaluate a definite integral?

If f is continuous on a finite interval [a,b] and F is an antiderivative on that interval, ∫ₐᵇ f(x) dx=F(b)−F(a). Improper integrals need separate convergence analysis.

How do scalar and vector quantities differ in mechanics?

A scalar is specified by a value and unit; a vector also has direction. Mass is scalar and displacement is vector; distance and displacement are not interchangeable.

How do assignment and equality comparison differ in Python 3?

`x = 3` binds x to the integer 3; `x == 3` compares values and is True here. `is` tests object identity rather than value equality.

What makes f continuous at an interior point a?

The value f(a) is defined, the limit lim(x→a) f(x) exists, and that limit equals f(a). At a domain endpoint, use the limit from within the domain.

When does a finite two-sided limit lim(x→a) f(x)=L exist?

Both one-sided limits must exist and equal L. The value f(a) may differ from L or be undefined; the limit describes nearby behavior.

What does a definite integral measure for integrable f on [a,b], with a<b?

∫ₐᵇ f(x) dx gives signed accumulation. Regions below the x-axis subtract; geometric area between the graph and axis is ∫ₐᵇ |f(x)| dx.

How do velocity and speed differ in a given reference frame?

Average velocity is displacement divided by elapsed time. Instantaneous velocity is the time derivative of position; instantaneous speed is its magnitude. Average speed uses total distance, not displacement magnitude.

What does a full-headed curved arrow show in a polar mechanism?

Movement of an electron pair from a lone pair or bond toward its destination. The arrow starts at electrons and describes electron redistribution, not an atom's trajectory.

How do nucleophiles and electrophiles differ?

A nucleophile donates an electron pair to form a bond; an electrophile accepts it. These roles do not require every nucleophile to be negative or every electrophile to be positive.

Which functional groups distinguish CH3CH2OH and CH3COOH?

CH3CH2OH is an alcohol, with a hydroxyl group on saturated carbon. CH3COOH contains a carboxyl group, −C(=O)OH; its OH is part of that acid group, not an ordinary alcohol group.

What assumptions support O(log n) time for binary search?

An already sorted random-access array, constant-time indexing and comparisons, and halving the remaining search range at each step. Sorting cost is excluded; finding first or last duplicates needs suitable extra logic.

How is the derivative f′(a) defined at an interior point?

When the finite limit exists, f′(a)=lim(h→0)[f(a+h)−f(a)]/h. It gives a local rate of change; its units are units of f divided by units of x.

Why do Newton's third-law forces not cancel on one body's force diagram?

The equal and opposite interaction forces act on different bodies. A force diagram for one body includes forces on that body, not the reaction forces it exerts on others.

Can acceleration occur at constant speed?

Yes. Acceleration is the time derivative of velocity, so changing direction can produce acceleration even at constant speed, as in uniform circular motion.

How does an algorithm differ from its implementation?

An algorithm specifies a procedure for a task with defined inputs and desired outputs. An implementation expresses it in a particular language; correctness and efficiency still require checking.

What is the work of a constant force over displacement Δr?

W=F·Δr=|F||Δr|cosθ, where θ is the angle between force and displacement. Work can be positive, negative or zero; varying forces require integration.

What does T(n)=O(n) say about an algorithm's cost?

For sufficiently large n, the chosen nonnegative cost T(n) is bounded above by Cn for some constant C. Specify input size, cost model and case; this need not be a tight bound or measured seconds.

How do constitutional isomers and stereoisomers differ?

Both share a molecular formula. Constitutional isomers differ in connectivity; stereoisomers share connectivity but differ in spatial arrangement. Resonance drawings are not separate isomers.

Must a Python 3 loop terminate?

No. A `while` loop can continue indefinitely if its condition stays true and nothing exits it. A `for` loop reaches exhaustion if its iteration stays finite and each iterator step and loop-body execution completes; it may also exit earlier.

When does Newton's second law take the form ΣF=ma?

In Newtonian mechanics with constant mass and an inertial frame, ΣF=ma is a vector relation. Use net external force and particle acceleration, or center-of-mass acceleration for a system.

What distinguishes a Brønsted–Lowry acid from a base?

An acid donates a proton; a base accepts one in the reaction considered. Acid strength is not the same as concentration or pH.

When is a particle's mechanical energy K+U conserved?

With a time-independent conservative potential and no work by other forces, K+U is constant. Nonconservative work can change mechanical energy without violating total-energy conservation.

What can dimensional analysis establish about an equation?

Both sides must have matching physical dimensions, and added terms must be compatible. This can reveal errors, but matching dimensions alone cannot prove the model or its numerical coefficient.

How does an if statement choose branches in Python 3?

It tests the condition's truth value; zero and empty lists are false, for example. An `elif` condition is considered only when earlier conditions were false.

What do resonance structures represent?

Different valid electron arrangements with the same atom positions and connectivity. The real species is a resonance hybrid, not molecules switching between drawings; the contributors need not have equal weight.

What characterizes an SN2 reaction at a stereogenic carbon?

Bond formation and bond breaking occur in one concerted step with backside attack. Configuration at the reacting carbon inverts; this does not guarantee that an R label becomes S.

How is formal charge calculated in a Lewis structure?

Formal charge = valence electrons − nonbonding electrons − half the bonding electrons. For nitrogen in NH4+, 5−0−4=+1. Formal charge is not a partial-charge measurement.

What is the power rule for a positive integer n?

d(x^n)/dx = n x^(n−1). For example, d(x³)/dx=3x². A constant has derivative zero; extensions to other powers require attention to the domain.

What is the product rule for differentiable functions f and g?

(fg)′=f′g+fg′, with all terms evaluated at the same x. It is not f′g′; for example, the derivative of x²x³ is 5x⁴.

Which integer indices access items of a Python 3 list of length n?

For n>0, valid indices are 0 to n−1 and −n to −1. An empty list has no valid item index; an out-of-range item index raises `IndexError`.

What is the chain rule for differentiable compositions?

If g is differentiable at x and f at g(x), d[f(g(x))]/dx = f′(g(x))g′(x). For example, d[(x²+1)³]/dx=6x(x²+1)².

How do parameters, arguments and return values differ in Python 3?

In `def f(x): return x+1`, x is a parameter; in `f(2)`, 2 is an argument and the returned value is 3. Printing output is different from returning a value.

STEM Foundations: Organic Chemistry, Mechanics, Calculus & Computing

32 introductory cards on selected foundations of organic chemistry, classical mechanics, single-variable calculus and basic computing, with Python 3 examples. Supports course review and problem practice; not a complete university STEM course.

Level: intermediate
Languages: en → en
Cards: 32
Markdown: STEM Foundations: Organic Chemistry, Mechanics, Calculus & Computing (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How does the fundamental theorem evaluate a definite integral?	If f is continuous on a finite interval [a,b] and F is an antiderivative on that interval, ∫ₐᵇ f(x) dx=F(b)−F(a). Improper integrals need separate convergence analysis.
2	How do scalar and vector quantities differ in mechanics?	A scalar is specified by a value and unit; a vector also has direction. Mass is scalar and displacement is vector; distance and displacement are not interchangeable.
3	How do assignment and equality comparison differ in Python 3?	`x = 3` binds x to the integer 3; `x == 3` compares values and is True here. `is` tests object identity rather than value equality.
4	What makes f continuous at an interior point a?	The value f(a) is defined, the limit lim(x→a) f(x) exists, and that limit equals f(a). At a domain endpoint, use the limit from within the domain.
5	When does a finite two-sided limit lim(x→a) f(x)=L exist?	Both one-sided limits must exist and equal L. The value f(a) may differ from L or be undefined; the limit describes nearby behavior.
6	What does a definite integral measure for integrable f on [a,b], with a<b?	∫ₐᵇ f(x) dx gives signed accumulation. Regions below the x-axis subtract; geometric area between the graph and axis is ∫ₐᵇ \|f(x)\| dx.
7	How do velocity and speed differ in a given reference frame?	Average velocity is displacement divided by elapsed time. Instantaneous velocity is the time derivative of position; instantaneous speed is its magnitude. Average speed uses total distance, not displacement magnitude.
8	What does a full-headed curved arrow show in a polar mechanism?	Movement of an electron pair from a lone pair or bond toward its destination. The arrow starts at electrons and describes electron redistribution, not an atom's trajectory.
9	How do nucleophiles and electrophiles differ?	A nucleophile donates an electron pair to form a bond; an electrophile accepts it. These roles do not require every nucleophile to be negative or every electrophile to be positive.
10	Which functional groups distinguish CH3CH2OH and CH3COOH?	CH3CH2OH is an alcohol, with a hydroxyl group on saturated carbon. CH3COOH contains a carboxyl group, −C(=O)OH; its OH is part of that acid group, not an ordinary alcohol group.
11	What assumptions support O(log n) time for binary search?	An already sorted random-access array, constant-time indexing and comparisons, and halving the remaining search range at each step. Sorting cost is excluded; finding first or last duplicates needs suitable extra logic.
12	How is the derivative f′(a) defined at an interior point?	When the finite limit exists, f′(a)=lim(h→0)[f(a+h)−f(a)]/h. It gives a local rate of change; its units are units of f divided by units of x.
13	Why do Newton's third-law forces not cancel on one body's force diagram?	The equal and opposite interaction forces act on different bodies. A force diagram for one body includes forces on that body, not the reaction forces it exerts on others.
14	Can acceleration occur at constant speed?	Yes. Acceleration is the time derivative of velocity, so changing direction can produce acceleration even at constant speed, as in uniform circular motion.
15	How does an algorithm differ from its implementation?	An algorithm specifies a procedure for a task with defined inputs and desired outputs. An implementation expresses it in a particular language; correctness and efficiency still require checking.
16	What is the work of a constant force over displacement Δr?	W=F·Δr=\|F\|\|Δr\|cosθ, where θ is the angle between force and displacement. Work can be positive, negative or zero; varying forces require integration.
17	What does T(n)=O(n) say about an algorithm's cost?	For sufficiently large n, the chosen nonnegative cost T(n) is bounded above by Cn for some constant C. Specify input size, cost model and case; this need not be a tight bound or measured seconds.
18	How do constitutional isomers and stereoisomers differ?	Both share a molecular formula. Constitutional isomers differ in connectivity; stereoisomers share connectivity but differ in spatial arrangement. Resonance drawings are not separate isomers.
19	Must a Python 3 loop terminate?	No. A `while` loop can continue indefinitely if its condition stays true and nothing exits it. A `for` loop reaches exhaustion if its iteration stays finite and each iterator step and loop-body execution completes; it may also exit earlier.
20	When does Newton's second law take the form ΣF=ma?	In Newtonian mechanics with constant mass and an inertial frame, ΣF=ma is a vector relation. Use net external force and particle acceleration, or center-of-mass acceleration for a system.
21	What distinguishes a Brønsted–Lowry acid from a base?	An acid donates a proton; a base accepts one in the reaction considered. Acid strength is not the same as concentration or pH.
22	When is a particle's mechanical energy K+U conserved?	With a time-independent conservative potential and no work by other forces, K+U is constant. Nonconservative work can change mechanical energy without violating total-energy conservation.
23	What can dimensional analysis establish about an equation?	Both sides must have matching physical dimensions, and added terms must be compatible. This can reveal errors, but matching dimensions alone cannot prove the model or its numerical coefficient.
24	How does an if statement choose branches in Python 3?	It tests the condition's truth value; zero and empty lists are false, for example. An `elif` condition is considered only when earlier conditions were false.
25	What do resonance structures represent?	Different valid electron arrangements with the same atom positions and connectivity. The real species is a resonance hybrid, not molecules switching between drawings; the contributors need not have equal weight.
26	What characterizes an SN2 reaction at a stereogenic carbon?	Bond formation and bond breaking occur in one concerted step with backside attack. Configuration at the reacting carbon inverts; this does not guarantee that an R label becomes S.
27	How is formal charge calculated in a Lewis structure?	Formal charge = valence electrons − nonbonding electrons − half the bonding electrons. For nitrogen in NH4+, 5−0−4=+1. Formal charge is not a partial-charge measurement.
28	What is the power rule for a positive integer n?	d(x^n)/dx = n x^(n−1). For example, d(x³)/dx=3x². A constant has derivative zero; extensions to other powers require attention to the domain.
29	What is the product rule for differentiable functions f and g?	(fg)′=f′g+fg′, with all terms evaluated at the same x. It is not f′g′; for example, the derivative of x²x³ is 5x⁴.
30	Which integer indices access items of a Python 3 list of length n?	For n>0, valid indices are 0 to n−1 and −n to −1. An empty list has no valid item index; an out-of-range item index raises `IndexError`.
31	What is the chain rule for differentiable compositions?	If g is differentiable at x and f at g(x), d[f(g(x))]/dx = f′(g(x))g′(x). For example, d[(x²+1)³]/dx=6x(x²+1)².
32	How do parameters, arguments and return values differ in Python 3?	In `def f(x): return x+1`, x is a parameter; in `f(2)`, 2 is an argument and the returned value is 3. Printing output is different from returning a value.