Calculus Core Concepts

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Questions and Answers

A function's concavity can be determined using which of the following?

  • The second derivative, $f''(x)$ (correct)
  • The limit of the function as x approaches infinity
  • The first derivative, $f'(x)$
  • The integral of the function, $∫f(x)dx$

Which of the following is an application of integration?

  • Finding the instantaneous rate of change of a function.
  • Determining the slope of a tangent line at a specific point.
  • Finding critical points of a function.
  • Calculating the area between curves. (correct)

According to the Fundamental Theorem of Calculus, what is the relationship between differentiation and integration?

  • Differentiation is the application of integration.
  • Integration is the application of differentiation.
  • They are unrelated processes.
  • They are inverse processes of each other. (correct)

What does the limit of a function describe?

<p>The value that a function approaches as the input approaches some value. (A)</p> Signup and view all the answers

If $\lim_{n \to \infty} a_n \neq 0$, what can be concluded about the series $\sum a_n$?

<p>The series diverges. (D)</p> Signup and view all the answers

When should Integration by Parts be considered as a method for solving an integral?

<p>When integrating the product of two functions. (C)</p> Signup and view all the answers

Which of the following is the correct application of the power rule for differentiation?

<p>$d/dx (x^n) = nx^{n-1}$ (A)</p> Signup and view all the answers

What does the derivative of a function at a point represent graphically?

<p>The slope of the tangent line at that point. (C)</p> Signup and view all the answers

In the context of infinite series, what does it mean for a series to converge?

<p>The sequence of its partial sums approaches a finite limit. (D)</p> Signup and view all the answers

What is the purpose of using trigonometric substitution in integration?

<p>To simplify integrals involving square roots of quadratic expressions. (D)</p> Signup and view all the answers

Flashcards

What is a Limit?

Value a function approaches as the input approaches a certain value.

What is a Derivative?

Measures the instantaneous rate of change of a function.

Power Rule (Derivative)

d/dx (x^n) = nx^(n-1)

Product Rule (Derivative)

d/dx (uv) = u'v + uv'

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Quotient Rule (Derivative)

d/dx (u/v) = (u'v - uv') / v^2

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Chain Rule (Derivative)

d/dx [f(g(x))] = f'(g(x)) * g'(x)

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What is Integration?

The inverse process of differentiation, finding the area under a curve.

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What is an Indefinite Integral?

∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration

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What is a Definite Integral?

∫[a to b] f(x) dx = F(b) - F(a), where F(x) is an antiderivative of f(x)

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Fundamental Theorem of Calculus (Part 1)

If f is continuous on [a, b] and F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x)

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