مجموعة من الأسئلة على مفاهيم وظيفة في حساب التفاضل والتكامل

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 .