92 free flashcards covering all 5 units. Study key concepts, terms, and exam-relevant topics.
Define a convergent power series and give the radius of convergence formula.
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.
What is a slope field and how does it help solve differential equations?
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.
Write the first four nonzero terms of the Maclaurin series for eˣ.
eˣ = 1 + x + x²/2! + x³/3! + …
Being able to write Taylor series helps answer function approximation questions on the exam.
When is the Ratio Test used and what does a limit L<1 imply?
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.
Compare the Alternating Series Test to the Direct Comparison Test.
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.
What is the epsilon‑delta definition of \(\lim_{x\to a} f(x) = L\)?
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.
How does the right‑hand limit \(\lim_{x\to a^+} f(x)\) differ from the two‑sided limit?
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.
State the condition for a function to be continuous at point a.
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.
How do you find horizontal asymptotes using limits at infinity?
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.
What is the Squeeze Theorem and when is it used?
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.
Define Limit
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.
One-Sided Limit
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.
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