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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?
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 $x$?
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?
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?
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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?
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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 $f$?
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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?
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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 $[ rac{ heta}{4}, rac{3 heta}{4}]$?
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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 $[0, 16]$?
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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?
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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?
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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?
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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
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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.
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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)
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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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Description
Test your knowledge of the applications of derivatives in calculus. This quiz covers key concepts from Unit 5 and addresses true/false statements and computation of derivatives. Be sure to work through each question and review your answers for a comprehensive understanding.
