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Math 271 Calc 1 > Final Exam Calc 1 > Flashcards

Final Exam Calc 1 Flashcards

1
Q

What is L’Hôpital’s Rule, and when can it be used?

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2
Q

How do you calculate net change using velocity?

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3
Q

What is the substitution rule for integration?

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4
Q

What are the two parts of the Fundamental Theorem of Calculus?

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5
Q

How do you evaluate a definite integral?

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6
Q

What methods approximate the area under a curve?

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Left Riemann Sum: Use left endpoints of intervals.
Right Riemann Sum: Use right endpoints.
Midpoint Rule: Use midpoints of intervals.
Trapezoidal Rule: Average left and right sums.

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7
Q

What is an antiderivative, and how is it related to a definite integral?

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8
Q

What is the formula for linear approximation?

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9
Q

How do you solve optimization problems?

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10
Q

What does the derivative tell us about a function?

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11
Q

State the Mean Value Theorem.

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12
Q

How do you find maxima and minima?

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13
Q

How do you solve related rates problems?

A
  1. Identify variables and rates.
  2. Write an equation relating the variables.
  3. Differentiate implicitly with respect to time (𝑡).
  4. Substitute known values and solve.
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14
Q

What are the derivatives of inverse trig functions?

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15
Q

What are the derivatives of exponential and logarithmic functions?

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16
Q

How do you perform implicit differentiation?

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17
Q

What is the chain rule?

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18
Q

State the product and quotient rules.

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19
Q

What are the conditions for continuity at 𝑥 = 𝑐?

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20
Q

What happens to a function as x→∞ or x → c where f(x)→∞?

A

Limits at infinity: Evaluate horizontal asymptotes.
Infinite limits: Describe vertical asymptotes.

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21
Q

What techniques are used to compute limits?

A

Direct substitution.
Factoring.
Rationalizing.
L’Hôpital’s Rule.

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22
Q

What is the definition of a limit?

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23
Q

When should you use L’Hôpital’s Rule?

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24
Q

When should you use the net change formula?

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Use the net change formula when you need to find the total change of a quantity over time, given its rate of change (e.g., velocity to find displacement).

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25
Q

When should you use the substitution rule for integration?

A

Use substitution when the integrand includes a composite function f(g(x))g′(x), and you can set u=g(x) to simplify the integral.

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26
Q

When should you use the Fundamental Theorem of Calculus?

A

Use Part 1 when differentiating a definite integral with a variable as an upper limit.
Use Part 2 when evaluating a definite integral by finding the antiderivative.

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27
Q

What is Part 1 of the Fundamental Theorem of Calculus?

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28
Q

What is Part 2 of the Fundamental Theorem of Calculus?

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29
Q

How are Part 1 and Part 2 of the Fundamental Theorem of Calculus related?

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30
Q

When do you use Part 1 of the Fundamental Theorem of Calculus?

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31
Q

When do you use Part 2 of the Fundamental Theorem of Calculus?

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32
Q

When should you evaluate a definite integral?

A

Use a definite integral when you need the exact value of the area under a curve over a specific interval [a,b].

33
Q

When should you approximate the area under a curve instead of integrating?

A

Use approximation methods (e.g., Riemann sums, trapezoidal rule) when the function is too complex to integrate directly, or you’re working with data points instead of a formula.

34
Q

When should you find an antiderivative?

A

Find antiderivatives when solving indefinite integrals or when determining the original function from a derivative.

35
Q

When should you use linear approximation?

A

Use linear approximation to estimate the value of a function near a point 𝑎, especially when finding the exact value is difficult.

36
Q

When should you use optimization techniques?

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Use optimization when trying to maximize or minimize a quantity, such as cost, area, or volume, given a set of constraints.

37
Q

When should you use a derivative to analyze a function?

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Use the derivative to determine the function’s increasing or decreasing behavior, find critical points, and identify local maxima or minima.

38
Q

When should you apply the Mean Value Theorem?

A

Use the Mean Value Theorem when a function is continuous on [a,b] and differentiable on (a,b) to guarantee a point where the instantaneous rate of change equals the average rate of change.

39
Q

When should you find maxima and minima?

A

Find maxima and minima when solving for critical points in optimization problems or analyzing a function’s behavior.

40
Q

When should you solve related rates problems?

A

Use related rates when two or more variables are changing with respect to time, and you need to find the rate of change of one variable in terms of others.

41
Q

When should you use the derivatives of inverse trig functions?

A

Use these derivatives when the function involves inverse trigonometric expressions like arcsin(x), arccos(x), or arctan(x).

42
Q

When should you use the rules for exponential and logarithmic derivatives?

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43
Q

When should you use implicit differentiation?

A

Use implicit differentiation when the function is not explicitly solved for 𝑦, or when 𝑦 is defined implicitly by an equation.

43
Q

When should you use the chain rule?

A

Use the chain rule when differentiating a composite function f(g(x)), where one function is nested inside another.

44
Q

When should you use the product and quotient rules?

A

Use the product rule when differentiating the product of two functions.
Use the quotient rule when differentiating the division of two functions.

45
Q

When should you check for continuity?

A

Check for continuity when determining if a function is smooth and unbroken over an interval or at a specific point.

46
Q

When should you evaluate limits at infinity or infinite limits?

A

Use limits at infinity to determine horizontal asymptotes.
Use infinite limits to analyze vertical asymptotes or unbounded behavior.

47
Q

When should you apply different techniques for computing limits?

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48
Q

When should you use the definition of a limit?

A

Use the formal definition to prove a limit exists, especially for rigorous proofs or theoretical problems.

49
Q

When should you apply the intuitive idea of a limit?

A

Use the intuitive idea of a limit to estimate or understand the behavior of a function as 𝑥 approaches a point or infinity.

50
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51
Q

What are the 3 steps to test for continuity at a point x=c?

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52
Q

What are the types of discontinuity in a function?

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53
Q

How do you check continuity for a piecewise function?

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54
Q

How do you find the limit of a piecewise function at a boundary?

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55
Q

How can you identify a discontinuity in a function’s graph?

A

Look for points where:

  1. The graph has a hole (removable discontinuity).
  2. The graph has a break or jump.
  3. The graph has a vertical asymptote (infinite discontinuity).
  4. The function oscillates wildly near a point.
56
Q

How are continuity and limits related?

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57
Q

How can you graphically identify the type of discontinuity in a piecewise function?

A
  1. Hole: A missing point in the graph (removable).
  2. Break/Jump: A sudden shift in the graph.
  3. Vertical Asymptote: The graph shoots up or down to infinity.
58
Q

How do you find the slope of the tangent line and the equation of the tangent line to a curve at a point?

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59
Q

What are the intervals a Polynomial function is continuous?

A

A Polynomial Function is defined for all values of x. The function is continuous on the interval (-infinity, infinity)

60
Q

How do you use limits to find the slope of the tangent line to a curve at a point?

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61
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62
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63
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64
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Math 271 Calc 1 (13 decks)

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