Podcast
Questions and Answers
Which intervals indicate that the graph of the function $f(x) = x^4 - 16x^2 + 8$ is concave up?
Which intervals indicate that the graph of the function $f(x) = x^4 - 16x^2 + 8$ is concave up?
- For $x < -4$ and $x > 4$
- For $x < -2$ and $x > 2$
- For $-2 < x < 2$ (correct)
- For $x = 0$ only
How can it be shown that the function $f(x) = an(x)$ is increasing for all defined $x$?
How can it be shown that the function $f(x) = an(x)$ is increasing for all defined $x$?
- By noting that its second derivative is positive
- By evaluating $f(x)$ at several points
- By confirming all local minima to be less than local maxima
- By finding that its first derivative, $f'(x) = rac{1}{ ext{cos}^2(x)}$, is always positive where defined (correct)
What can be inferred about the graph of the original function $y = f(x)$ at $x = a$ where $f'(x)$ has a horizontal tangent?
What can be inferred about the graph of the original function $y = f(x)$ at $x = a$ where $f'(x)$ has a horizontal tangent?
- The graph is transitioning from increasing to decreasing
- The function switches from concave up to concave down
- The function has an inflection point at $x = a$ (correct)
- The function is instantaneously constant at $x = a$
What can be concluded about the original function $f(x)$ at $x = b$ if $f'(x)$ is positive and increasing?
What can be concluded about the original function $f(x)$ at $x = b$ if $f'(x)$ is positive and increasing?
If a function $f$ is continuous on the closed interval $[a, b]$, which statement is true regarding its derivative?
If a function $f$ is continuous on the closed interval $[a, b]$, which statement is true regarding its derivative?
Given that $f(a) = 5$ and $f(b) = 5$, what can be concluded about the graph of $f$?
Given that $f(a) = 5$ and $f(b) = 5$, what can be concluded about the graph of $f$?
Which of the following statements about a local maximum $g'(c)$ of $g'(x)$ is true?
Which of the following statements about a local maximum $g'(c)$ of $g'(x)$ is true?
Rolle's Theorem applies to which of the following functions on the interval $[ rac{ heta}{4}, rac{3 heta}{4}]$?
Rolle's Theorem applies to which of the following functions on the interval $[ rac{ heta}{4}, rac{3 heta}{4}]$?
For the function $f(x) = 2 ext{sqrtn}(x)$, what is the derivative $f'(c)$ given the average rate of change on the interval $[0, 16]$?
For the function $f(x) = 2 ext{sqrtn}(x)$, what is the derivative $f'(c)$ given the average rate of change on the interval $[0, 16]$?
When considering a particle's movement graph from $0$ to $11$, on which intervals is the acceleration likely to be negative?
When considering a particle's movement graph from $0$ to $11$, on which intervals is the acceleration likely to be negative?
What is the condition for the existence of a minimum or maximum value of a function on a closed interval?
What is the condition for the existence of a minimum or maximum value of a function on a closed interval?
In a situation where a particle's position graph crosses the horizontal axis, what does it imply?
In a situation where a particle's position graph crosses the horizontal axis, what does it imply?
Flashcards
Rolle's Theorem
Rolle's Theorem
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one value c in (a, b) such that f'(c) = 0.
Mean Value Theorem
Mean Value Theorem
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one value c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
Local Maximum/Minimum of f'(x)
Local Maximum/Minimum of f'(x)
A critical point of f'(x) where f''(x) changes sign.
Continuity on closed interval
Continuity on closed interval
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Differentiable
Differentiable
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Average Rate of Change
Average Rate of Change
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Optimization Problem
Optimization Problem
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Acceleration of a particle (motion)
Acceleration of a particle (motion)
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Speeding up
Speeding up
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Concavity of f(x)
Concavity of f(x)
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Absolute maximum of f(x)
Absolute maximum of f(x)
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Increasing Function
Increasing Function
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Limit of a Function
Limit of a Function
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Graph of f'(x) and f(x)
Graph of f'(x) and f(x)
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Study Notes
Practice Test - Unit 5
- Applications of Derivatives
- Instructions: Answer every question. Show work. Review solutions.
- Section 4.7 and Unit 5 ideas in connection with 4.2
- Question 1: Determine if each statement is true or false.
- I. If a function f is continuous on [a, b], then there is at least one value c such that f'(c) = (f(b)-f(a))/(b-a).
- True/False
- II. If f(a) = 5 and f(b) = 5, there's at least one point in (a, b) where the graph of f has a horizontal tangent line.
- True/False
- III. If g'(c) is a local maximum value of g'(x), then the graph of g has an inflection point at x = c.
- True/False
- IV. Any function continuous on an open interval necessarily has a greatest and least value on a corresponding closed interval.
- True/False
- I. If a function f is continuous on [a, b], then there is at least one value c such that f'(c) = (f(b)-f(a))/(b-a).
- Question 2: For f(x) = 2√x, find all values c such that f'(c) equals the average rate of change on [0, 16].
- Question 3: Calculator question. What point on y = (x-5)² is closest to the origin?
- Question 4: Explain why Rolle's theorem applies to f(x) = cos(2x) on the closed interval ≤ x ≤ 4.
- Question 5: The position of a particle is graphed (0 ≤ t ≤ 11).
- (a) On what intervals is the acceleration negative? Is it ever positive? Justify your answer.
- (b) When is the object speeding up? Justify your answer.
- Question 6: Find intervals where the graph of f(x) = (x^4/6) - (x²/16) + 8 is concave up and concave down.
- Question 7: What is the absolute maximum value of f(x) = eˣcos(x) on [0,1]?
- Question 8: Show that f(x) = tan(x) is increasing for all x where the function is defined.
- Question 9: Evaluate the limits.
- (a) lim x→2 (1 + cos(2x))/ln(2x)
- (b) lim x→0 (1 - e⁻ˣ)/sin(x)
- Question 10: The derivative of a twice-differentiable function f is graphed.
- (a) What can be inferred about the graph of the original function f(x) at x = a and state the reason?
- (b) What can be inferred about the graph of the original function f(x) at x = b and state the reason?
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