CLEP CLEP Calculus Flashcards

A power series ∑aₙ(x−c)ⁿ converges for |x−c|<R, diverges for |x−c|>R. The radius R = 1/limsupₙ|aₙ|^{1/n}.

Knowing how to calculate R lets you determine where Taylor/Maclaurin series apply in the exam.

A slope field graphs dy/dx = f(x,y) at discrete points; it visualizes solution curves without explicit integration.

Slope fields show qualitative behavior of solutions, an important concept for the differential equations section of the test.

eˣ = 1 + x + x²/2! + x³/3! + …

Being able to write Taylor series helps answer function approximation questions on the exam.

Use it for ∑aₙ where lim|a_{n+1}/aₙ| = L; if L<1, the series converges absolutely.

The Ratio Test is a popular convergence test frequently asked on the exam.

Alternating test applies to ∑(−1)^{n}bₙ with decreasing bₙ→0 ⇒ convergence; Direct comparison test matches bₙ with a known convergent or divergent series to deduce behavior.

Both are key tools for deciding series convergence on the CLEP test, so spotting which one fits saves time.

For every ε>0 there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε.

This formal definition underpins limit theorems and continuity checks on the exam.

The right‑hand limit considers only x>a approaching a; the two‑sided limit requires both sides to agree.

One‑sided limits are required to test continuity at discontinuities like holes or jumps.

f(a) exists, \(\lim_{x\to a}f(x)\) exists, and \(\lim_{x\to a}f(x)=f(a)\).

Continuity ensures the Intermediate Value Theorem applies—a core exam concept.

Compute \(\lim_{x\to±∞}f(x)\); if finite, that value is a horizontal asymptote.

Recognizing asymptotes is vital for evaluating end‑behavior limits and graph sketching.

If g(x)≤f(x)≤h(x) and \(\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L\), then \(\lim_{x\to a}f(x)=L\). Use for indeterminate forms.

The theorem is a powerful tool for tricky limits that aren’t solvable by algebraic simplification.

A value a function approaches as x approaches a certain point

Understanding limits is crucial for the CLEP Calculus exam as it forms the foundation of calculus concepts.

Limit as x approaches from one side

Recognizing one-sided limits is essential for evaluating functions with different behaviors on each side of a point.

Function with no gaps or jumps

Identifying continuous functions is vital for applying the Intermediate Value Theorem (IVT) and evaluating limits.

Limit as x approaches infinity or negative infinity

Evaluating limits at infinity helps in understanding the behavior of functions as x becomes very large or very small.

Method for evaluating limits using inequalities

The Squeeze Theorem is a key technique for evaluating limits, especially when other methods fail, and is frequently tested on the CLEP Calculus exam.

Limit as x approaches from left or right

Understanding one-sided limits is crucial for evaluating limits of functions with discontinuities or infinite limits.

Continuity implies IVT, but not vice versa

This distinction is vital for determining the existence of roots or extreme values of a function within a given interval.

Limit as x approaches infinity or negative infinity

Limits at infinity help analyze the behavior of functions as x becomes arbitrarily large, a key concept in calculus.

Limits can be found by bounding functions

The Squeeze Theorem is a fundamental technique for evaluating limits, especially when dealing with trigonometric or rational functions.

Removable: limit exists, Essential: limit does not exist

Understanding the types of discontinuities is essential for evaluating limits and determining the continuity of functions, a fundamental concept in calculus.