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WorksheetsΕρωτήσεις Θεωρίας
Total questions: 61
Worksheet time: 2hrs 21mins
What do we call a real function with domain A?
What is called the set of values of a function f with domain A?
What do we call the graphical representation of a function?
How does the graphical representation of the functions f and -f arise with the help of the graphical representation of f?
When are two functions f and g equal?
How are the operations of functions defined?
Sum fg +
Difference fg -
Product fg
Quotient f/g
What do we call the composition of f?
What do we call the composition of f with g?
When is a function said to be strictly increasing, strictly decreasing, and strictly monotonic in an interval Δ of its domain?
When do we say that a function has a maximum and when a minimum?
When it has a global maximum at a point
When it has a global minimum at a point
When it is increasing
When it is decreasing
When is a function said to be 1-1? What are the criteria for a function to be 1-1?
How is the inverse function f^-1 of a function f defined and what do you know about their graphs?
Explain why the graphs of the functions f and f^-1 are symmetric with respect to the line y=x that bisects the angles xOy and x'Oy'.
Give the meaning of the limit lim (x→0) f(x) = ?
What is the meaning of the limit as x approaches 0 of f(x)?
When is a limit well-defined?
What is the meaning of the lateral limits as x approaches 0 from the left and right?
What are the consequences if a function f is defined on a set of the form (α, β)?
What are the consequences if a function f is defined on a set of the form (0, α)?
Ποια θεωρήματα ισχύουν για το όριο και τη διάταξη;
ΘΕΩΡΗΜΑ 1ο: Αν lim f(x) > 0, τότε f(x) > 0 κοντά στο 0.
ΘΕΩΡΗΜΑ 1ο: Αν lim f(x) < 0, τότε f(x) < 0 κοντά στο 0.
ΘΕΩΡΗΜΑ 2ο: Αν οι συναρτήσεις f, g έχουν όριο στο 0 και ισχύει f(x) ≤ g(x) κοντά στο 0, τότε lim f(x) ≤ lim g(x).
Ποιες είναι οι ιδιότητες των ορίων;
lim f(x) = lim g(x) εφόσον lim g(x) ≠ 0.
lim kf(x) = k lim f(x) για κάθε σταθερά k.
lim f(x) = lim f(x)
lim f(x) * g(x) = lim f(x) * lim g(x)
lim f(x) + g(x) = lim f(x) + lim g(x)
Να αποδείξετε ότι για οποιοδήποτε πολυώνυμο P(x), ισχύει lim (x→0) P(x) = 0.
Να αποδείξετε ότι για τα πολυώνυμα P(x), Q(x), με Q(x) ≠ 0, ισχύει lim (x→0) P(x)/Q(x) = 0.
Να διατυπώσετε το κριτήριο παρεμβολής.
Ποια είναι τα βασικά τριγωνομετρικά όρια στο 0 < x;
What are the basic trigonometric limits?
lim x->0 x = 0
lim x->0 x = 1
lim x->0 x = 0
lim x->0 x = 1
lim x->0 x = 0
How is the limit of the composition of two functions f, g at the point x0 defined?
Set u = g(x)
Calculate lim x->0 g(x)
Calculate lim u->0 f(u)
All of the above
What is the meaning of the limits lim x->∞ f(x) = L?
The values of f approach a real number L as x increases indefinitely.
The values of f approach infinity as x increases indefinitely.
The values of f approach negative infinity as x increases indefinitely.
None of the above
What is the meaning of the limits lim x->∞ f(x) = +∞?
The values of f increase indefinitely.
The values of f approach a real number.
The values of f decrease indefinitely.
None of the above
What are the properties of the infinite limit?
lim f(x) as x approaches 0 from the positive side equals +∞ if lim f(x) as x approaches 0 from the negative side equals +∞
lim f(x) as x approaches 0 from the positive side equals -∞ if lim f(x) as x approaches 0 from the negative side equals -∞
If lim f(x) as x approaches 0 equals +∞, then f(x) > 0 near x = 0, while if lim f(x) as x approaches 0 equals -∞, then f(x) < 0 near x = 0.
If lim f(x) as x approaches 0 equals +∞, then lim f(x) as x approaches 0 from the negative side equals -∞, while if lim f(x) as x approaches 0 equals -∞, then lim f(x) as x approaches 0 from the negative side equals +∞.
If lim f(x) as x approaches 0 equals +∞ or -∞, then lim f(x) as x approaches 0 equals +∞.
What are the limits of polynomial and rational functions at infinity?
For the polynomial function, the limit as x approaches +∞ is α.
For the polynomial function, the limit as x approaches -∞ is α.
For the rational function, the limit as x approaches +∞ is α/β.
For the rational function, the limit as x approaches -∞ is α/β.
What are the limits of the exponential and logarithmic functions at the ends of their domain?
If α > 1, then the limit as x approaches -∞ is 0.
If α > 1, then the limit as x approaches +∞ is +∞.
If 0 < α < 1, then the limit as x approaches -∞ is +∞.
If 0 < α < 1, then the limit as x approaches +∞ is 0.
What is called a sequence?
When does the sequence α_n have a limit?
What is called a sequence?
When does the sequence have a limit?
When do we say that a function f is continuous at x0 of its domain?
When do we say that a function is continuous?
When is a function continuous on (α, β)?
When is a function continuous on [α, β]?
If two functions f and g are continuous at x0, then which other functions defined through f and g are continuous at x0?
If two functions f and g are continuous at 0 x, what other functions defined through f and g are continuous at 0 x?
fg +
cf *
fg *
f g
| f |
State the Bolzano theorem and provide its geometric interpretation.
If a function f is continuous on a closed interval [α, β] and additionally, it holds that f(α) f(β) < 0, then there exists at least one x0 in (α, β) such that f(x0) = 0.
From the Bolzano theorem, it follows that if a function f is continuous on an interval Δ and does not become zero in it, then it is either positive for every x in Δ or negative for every x in Δ, meaning it maintains a sign in the interval Δ.
State the Intermediate Value Theorem, prove it and give its geometric interpretation.
Let f be a function defined on a closed interval [α, β]. If f is continuous on [α, β] and f(α) ≠ f(β), then for every number η between f(α) and f(β), there exists at least one x₀ in (α, β) such that f(x₀) = η.
What is true about the image of an interval Δ of a continuous and non-constant function?
The image f(Δ) of an interval Δ through a continuous and non-constant function f is an interval.
The image f(Δ) of an interval Δ through a continuous and non-constant function f is a point.
The image f(Δ) of an interval Δ through a continuous and non-constant function f is a set of discrete points.
State the maximum and minimum value theorem for a continuous function.
If f is a continuous function on [α, β], then f takes a maximum value M and a minimum value m on [α, β].
If f is a continuous function on [α, β], then f is constant on [α, β].
If f is a continuous function on [α, β], then f has no maximum or minimum values.
What do we define as the tangent of C f at the point of (0, A(x, f(x)))?
Provide the geometric interpretation of the tangent of C f at the point of (0, A(x, f(x)))?
When do we say that a function is differentiable at a point x0 in its domain?
If the limit exists and is a real number
If the function is continuous at that point
If the function has a maximum at that point
If the function is defined at that point
How is defined: a) the position function of a moving object, b) the average speed of a moving object, c) the instantaneous speed of a moving object at time t0 and d) the acceleration of a moving object at time t0?
What is the average speed of the object during the time interval from t0 to t?
What do we call the limit of the average speed as t approaches t0?
When does a mobile object move to the right and when to the left near t0?
When a mobile object moves to the right, the condition S(t0) - S(t) > 0 holds.
When a mobile object moves to the left, the condition S(t0) - S(t) < 0 holds.
What is called the slope of f at x0 and what is the equation of the tangent line of fC at x0?
The slope of f at x0 is the derivative f'(x0).
The equation of the tangent line is y - f(x0) = f'(x0)(x - x0).
Prove that if a function f is differentiable at a point x0 in its domain, then it is also continuous at that point.
When is a function f differentiable in its domain A?
When is a function f differentiable in an open interval (α, β) of its domain?
When is a function f differentiable in a closed interval [α, β] of its domain?
