مجموعة من الأسئلة على مفاهيم وظيفة في حساب التفاضل والتكامل
Question 1:
True or False. The two functions f and g defined by
f(x) = 3x + 3 for x real and g(t) = 3t + 3 for t real and positive
are equal?
Answer :
False. Two functions are equal if their rules are equal and their domains are the same.
Question 2:
If functions f and g have domains Df and Dg respectively, then the domain of f / g is given by
(A) the union of Df and Dg
(B) the intersection of Df and Dg
(C) the intersection of Df and Dg without the zeros of function g
(D) None of the above
Answer :
(C). Division by zero is not allowed in mathematics. Students tend to forget this point.
Question 3:
True or False. The graph of f(x) and that of f(x + 2) are the same
Answer :
False. The graph of f(x + 2) is that of f(x) shifted 2 units to the left.
Question 4:
Let the closed interval [a , b] be the domain of function f. The domain of f(x - 3) is given by
(A) the open interval (a , b)
(B) the closed interval [a , b]
(C) the closed interval [a - 3 , b - 3]
(D) the closed interval [a + 3 , b + 3]
Answer :
(D). The graph of f(x - 3) is that of f(x) shifted 3 units to the right. To shift the closed interval [a , b] to the right you need to add 3 units to the endpoints a and b of the interval.
Question 5:
Let the interval (a , +infinity) be the range of function f. The range of f(x) - 4 is given by
(A) the interval (a - 4 , +infinity)
(B) the interval (a + 4, +infinity)
(C) the interval (a, +infinity)
(C) None of the above
Answer :
(A). If the range of f is given by the interval (a , +infinity), we can write the following inequality
f(x) > a
add - 4 to both sides on the inequality to obtain
f(x) - 4 > a - 4
The last inequality suggests that the range of f(x) - 4 is (a - 4, +infinity)
Question 6:
True or False. The equation y = | x | , with y >= 0, represents y as a function of x.
Answer :
True.
Question 7:
True or False. The equation x = | y | , with x >= 0, represents y as a function of x.
Answer :
False. Solve for y to find that y = | x | or y = -| x |; for one value of the independent variable x we have two values of the dependent variable y.
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الأسئلة
التي صممت لمساعدتك في الحصول على الفهم العميق لخصائص الرسوم البيانية من
الوظائف التي تعتبر ذات أهمية كبرى في حساب التفاضل والتكامل.
Question 1:
True or False. The domain of a function is the set of all real values for which the function is real valued.
Answer :
True.
Question 2:
True or False. The sign of the first derivative of a given function f informs you on the interval(s) where f(x) is positive, negative or equal to zero.
Answer :
False.
The sign of the first derivative informs you on the interval(s) where f is increasing, decreasing or constant.
Question 3:
True or False. The sign of the second derivative of a given function f informs you on the concavity of the graph of f.
Answer :
True.
Question 4:
True or False. The horizontal asymptote to the graph of a given function f is determined by finding the limit, if it exists, of f(x) as x approaches 0.
Answer :
False.
A horizontal asymptote may be determined by finding the limit of f(x) as x approches + or - infinity (very large or very small values).
Question 5:
True or False. Any value of x that makes the denominator of rational function f equal to zero, represents a vertical asymptote to the graph of f.
Answer :
False.
Not always. Let f(x) = (x + 3) / (x 2 -9).
Factor the denominator and simplify to obtain f(x) = 1 / (x - 3)
Although x = - 3 makes the denominator equal to 0 there is no vertical asymptote at x = - 3; in fact there is a hole.
Question 6:
True or False. A horizontal asymptote may intersect the graph of the function.
Answer :
True.
Example: f(x) = sin x / x
Question 7:
True or False. The x intercepts of the graph of a function corresponds to the zeros of the function.
Answer :
True.
Question 8:
True or False. A graph cannot cut its vertical asymptote.
Answer :
True.
More references on calculus questions with answers and tutorials and problems .
Question 1:
True or False. The two functions f and g defined by
f(x) = 3x + 3 for x real and g(t) = 3t + 3 for t real and positive
are equal?
Answer :
False. Two functions are equal if their rules are equal and their domains are the same.
Question 2:
If functions f and g have domains Df and Dg respectively, then the domain of f / g is given by
(A) the union of Df and Dg
(B) the intersection of Df and Dg
(C) the intersection of Df and Dg without the zeros of function g
(D) None of the above
Answer :
(C). Division by zero is not allowed in mathematics. Students tend to forget this point.
Question 3:
True or False. The graph of f(x) and that of f(x + 2) are the same
Answer :
False. The graph of f(x + 2) is that of f(x) shifted 2 units to the left.
Question 4:
Let the closed interval [a , b] be the domain of function f. The domain of f(x - 3) is given by
(A) the open interval (a , b)
(B) the closed interval [a , b]
(C) the closed interval [a - 3 , b - 3]
(D) the closed interval [a + 3 , b + 3]
Answer :
(D). The graph of f(x - 3) is that of f(x) shifted 3 units to the right. To shift the closed interval [a , b] to the right you need to add 3 units to the endpoints a and b of the interval.
Question 5:
Let the interval (a , +infinity) be the range of function f. The range of f(x) - 4 is given by
(A) the interval (a - 4 , +infinity)
(B) the interval (a + 4, +infinity)
(C) the interval (a, +infinity)
(C) None of the above
Answer :
(A). If the range of f is given by the interval (a , +infinity), we can write the following inequality
f(x) > a
add - 4 to both sides on the inequality to obtain
f(x) - 4 > a - 4
The last inequality suggests that the range of f(x) - 4 is (a - 4, +infinity)
Question 6:
True or False. The equation y = | x | , with y >= 0, represents y as a function of x.
Answer :
True.
Question 7:
True or False. The equation x = | y | , with x >= 0, represents y as a function of x.
Answer :
False. Solve for y to find that y = | x | or y = -| x |; for one value of the independent variable x we have two values of the dependent variable y.
ـــــــــــــــــــــــ
الأسئلة
التي صممت لمساعدتك في الحصول على الفهم العميق لخصائص الرسوم البيانية من
الوظائف التي تعتبر ذات أهمية كبرى في حساب التفاضل والتكامل.
Question 1:
True or False. The domain of a function is the set of all real values for which the function is real valued.
Answer :
True.
Question 2:
True or False. The sign of the first derivative of a given function f informs you on the interval(s) where f(x) is positive, negative or equal to zero.
Answer :
False.
The sign of the first derivative informs you on the interval(s) where f is increasing, decreasing or constant.
Question 3:
True or False. The sign of the second derivative of a given function f informs you on the concavity of the graph of f.
Answer :
True.
Question 4:
True or False. The horizontal asymptote to the graph of a given function f is determined by finding the limit, if it exists, of f(x) as x approaches 0.
Answer :
False.
A horizontal asymptote may be determined by finding the limit of f(x) as x approches + or - infinity (very large or very small values).
Question 5:
True or False. Any value of x that makes the denominator of rational function f equal to zero, represents a vertical asymptote to the graph of f.
Answer :
False.
Not always. Let f(x) = (x + 3) / (x 2 -9).
Factor the denominator and simplify to obtain f(x) = 1 / (x - 3)
Although x = - 3 makes the denominator equal to 0 there is no vertical asymptote at x = - 3; in fact there is a hole.
Question 6:
True or False. A horizontal asymptote may intersect the graph of the function.
Answer :
True.
Example: f(x) = sin x / x
Question 7:
True or False. The x intercepts of the graph of a function corresponds to the zeros of the function.
Answer :
True.
Question 8:
True or False. A graph cannot cut its vertical asymptote.
Answer :
True.
More references on calculus questions with answers and tutorials and problems .
..asmaa2011-11-27, 5:47 pm