30 Questions
If a function f is differentiable at every point of its domain, what can be said about its continuity?
It is continuous at every point of its domain.
What can be said about the product function f(x).g(x) if f(x) is differentiable at x = a and g(x) is not differentiable at x = a?
It can be differentiable at x = a.
If a function f(x) is differentiable at x = a, what can be said about its left-hand and right-hand derivatives at x = a?
Both are equal and finite.
What is the condition for a function f(x) to be differentiable at a point P?
The function has a unique tangent at point P.
What is the right-hand derivative of f(x) at x = a, denoted by?
f'(a+0) or f'(a+)
What is the set of points where the function f(x) = 1 - e^(-x) is differentiable?
(-∞, 0) ∪ (0, ∞)
If f(x) and g(x) both are not differentiable at x = a, what can be said about the sum function f(x) + g(x)?
It can be differentiable at x = a.
What is the condition for a function f(x) to be differentiable everywhere?
The function has a unique tangent at every point.
What is the left-hand derivative of f(x) at x = a, denoted by?
f'(a-0) or f'(a-)
What can be said about the differentiability of the function f(x) = |x-1| at x = 1?
The function is not differentiable at x = 1.
What is the condition for a function f(x) to be differentiable at a point a?
lim (h → 0+) f(a + h) - f(a) / h = lim (h → 0-) f(a - h) - f(a) / h
What can be said about the function f(x) if it is differentiable at every point in an open interval (a, b)?
The function is continuous and has a unique tangent at every point in the interval.
Which of the following functions is everywhere differentiable?
f(x) = x^2
What is the relationship between the left-hand and right-hand derivatives of a function f(x) at a point a?
The left-hand derivative can be different from the right-hand derivative.
What is the geometric interpretation of the derivative of a function f(x) at a point a?
The slope of the tangent line at the point.
What can be said about the function f(x) = 1 + |x| at x = 0?
It is both differentiable and continuous at x = 0.
What is the derivative of the function f(x) = |x - 1| + |x - 3| at x = 2?
0
What can be said about the function f(x) = |x| at x = 0?
It is not differentiable at x = 0, but continuous.
What is the condition for a function f(x) to be differentiable at a point a?
The left-hand and right-hand derivatives of the function must be equal at x = a.
What can be said about the function f(x) = 1 - e^(-x) at x = 0?
It is both differentiable and continuous at x = 0.
What can be said about the function f(x) = |x - 1| + |x + 1|?
It is continuous at x = 1 and x = -1, but not differentiable.
What can be said about the function f(x) = (x-1)|x-3|x+2|+cos(|x|) at x = 2?
It is not differentiable at x = 2
What is the derivative of f(x) = |x - 1| + |x + 1| at x = 2?
2
What is the value of f(0) for the function f(x) = xe^|x| if x ≠ 0 and f(x) = 0 if x = 0?
0
At what points is the function f(x) = |x - 1| + |x + 1| not differentiable?
x = 1 and x = -1
What can be said about the function f(x) = xe^|x| at x = 0?
It is continuous but not differentiable at x = 0
What can be said about the function f(x) = 2x - 1 at x = 0?
It is both continuous and differentiable.
What is the limit of (f(x) - f(0))/h as h approaches 0 for the function f(x) = xe^|x| if x ≠ 0 and f(x) = 0 if x = 0?
0
What is the left-hand derivative of f(x) = |x - 1| + |x + 1| at x = 1?
0
What can be said about the function f(x) = (x-1)|x-3|x+2|+cos(|x|) for x < 1 or x > 2?
It is differentiable and continuous for all x
Understand the concept of differentiability of a function at a point, including the meaning and definition of differentiability. Learn how to identify and analyze differentiable functions.
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