Calculus I – Continuity, Differentiability & Derivative Rules

Flashcards cover continuity vs differentiability, corner points, practical relevance, spline concept, limit definition insights, derivative rules (constant, sum, power, product, quotient), derivative of e^x, and typical exam tips.

What are the three conditions for a function f to be continuous at x=a?

1) f(a) is defined. 2) The limit of f(x) as x→a exists. 3) The limit equals f(a).

Which of the three continuity conditions is usually the hardest to verify?

The existence of the limit (condition #2), because it requires left-hand and right-hand limits to agree.

What additional requirement must be met for a function to be differentiable at a point where it is already continuous?

The left-hand and right-hand derivative (slope) limits must agree; i.e., the slopes from both sides are equal.

Give a simple example of a function that is continuous but not differentiable at x=0.

f(x)=|x| (absolute value); it is continuous everywhere but has different slopes (–1 and +1) on each side of x=0, creating a corner point.

Why are ‘corner points’ acceptable for continuity but not for differentiability?

Because continuity only requires the function values to meet, whereas differentiability also requires matching slopes, which corners lack.

In engineering design (e.g., roller coasters), why is differentiability important?

Smoothness (matching slopes and often higher derivatives) prevents sudden changes that could cause mechanical failure or discomfort.

What is a spline in the context of data science or engineering?

A piecewise-defined, smooth function whose pieces meet with continuous value and usually continuous first (and even second) derivatives.

What does the phrase "When I say derivative, you say ___" emphasize?

Slope.

State the derivative of a linear function f(x)=mx+b.

f'(x)=m (the constant slope).

What is the derivative of a constant function c?

0 (its graph is a horizontal line with zero slope).

Constant Multiple Rule: If g(x)=k·f(x), what is g'(x)?

g'(x)=k·f'(x).

Sum (and Difference) Rule: If h(x)=f(x)+g(x), what is h'(x)?

h'(x)=f'(x)+g'(x) (subtract if it is a difference).

Power Rule: What is d/dx [x^n] for any real n?

d/dx [x^n]=n·x^{n−1}.

Use the Power Rule: d/dx [7x^5] = ?

35x^4.

Product Rule formula for P(x)=f(x)·g(x).

P'(x)=f'(x)·g(x)+g'(x)·f(x).

Quotient Rule formula for Q(x)=f(x)/g(x) (g(x)≠0).

Q'(x) = [f'(x)·g(x) − g'(x)·f(x)] / [g(x)]².

Compute the derivative using the Product Rule: d/dx [(3x^4)(2x−7)].

(12x^3)(2x−7) + (2)(3x^4) = 24x^4−84x^3.

Why does the Quotient Rule have the extra "denominator squared" term?

Division introduces an additional factor when applying limits; the denominator squared keeps the derivative expression properly scaled.

Derivative of the natural exponential function e^x.

d/dx [e^x] = e^x (it is its own derivative).

What special property makes e^x the “natural” exponential?

At any point, its slope equals its y-value, modeling processes where growth rate is proportional to current amount.

Give an example of a real-world process often modeled by e^x because its rate is proportional to its size.

Population growth (e.g., rabbit population increases faster when there are more rabbits).

If a function has a vertical tangent at x=a, is it differentiable there? Why or why not?

No; the slope approaches ±∞, so the derivative does not exist (it is unbounded).

Common first-exam freebie at UConn Calc I: "Provide a function continuous but not differentiable." What is the simple answer?

Draw or state f(x)=|x| and mention it has a corner point at x=0.

Why must the denominator be non-zero when using the Quotient Rule?

Because division by zero is undefined; differentiability requires the original function itself to be defined.

What upcoming calculus topic involves ensuring both first and second derivatives are continuous across piecewise segments?

Spline construction (smooth piecewise functions).