Differentiability

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If a function f is differentiable at every point of its domain, what can be said about its continuity?

It is continuous only at the midpoint of its domain.

It is continuous only at the end points of its domain.

It is not necessarily continuous at every point of its domain.

It is continuous at every point of its domain. (correct)

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. (correct)

It is always differentiable at x = a.

It is not defined at x = a.

It is never 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. (correct)

One is equal and finite, the other is infinite.

Both are unequal and infinite.

Both are equal but infinite.

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. Signup and view all the answers

What is the right-hand derivative of f(x) at x = a, denoted by?

f'(a+0) or f'(a+) Signup and view all the answers

What is the set of points where the function f(x) = 1 - e^(-x) is differentiable?

(-∞, 0) ∪ (0, ∞) Signup and view all the answers

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. Signup and view all the answers

What is the condition for a function f(x) to be differentiable everywhere?

The function has a unique tangent at every point. Signup and view all the answers

What is the left-hand derivative of f(x) at x = a, denoted by?

f'(a-0) or f'(a-) Signup and view all the answers

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. Signup and view all the answers

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 Signup and view all the answers

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. Signup and view all the answers

Which of the following functions is everywhere differentiable?

f(x) = x^2 Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

What can be said about the function f(x) = 1 + |x| at x = 0?

It is both differentiable and continuous at x = 0. Signup and view all the answers

What is the derivative of the function f(x) = |x - 1| + |x - 3| at x = 2?

0 Signup and view all the answers

What can be said about the function f(x) = |x| at x = 0?

It is not differentiable at x = 0, but continuous. Signup and view all the answers

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. Signup and view all the answers

What can be said about the function f(x) = 1 - e^(-x) at x = 0?

It is both differentiable and continuous at x = 0. Signup and view all the answers

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. Signup and view all the answers

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 Signup and view all the answers

What is the derivative of f(x) = |x - 1| + |x + 1| at x = 2?

2 Signup and view all the answers

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 Signup and view all the answers

At what points is the function f(x) = |x - 1| + |x + 1| not differentiable?

x = 1 and x = -1 Signup and view all the answers

What can be said about the function f(x) = xe^|x| at x = 0?

It is continuous but not differentiable at x = 0 Signup and view all the answers

What can be said about the function f(x) = 2x - 1 at x = 0?

It is both continuous and differentiable. Signup and view all the answers

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 Signup and view all the answers

What is the left-hand derivative of f(x) = |x - 1| + |x + 1| at x = 1?

0 Signup and view all the answers

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 Signup and view all the answers

Differentiability

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Questions and Answers

If a function f is differentiable at every point of its domain, what can be said about its continuity?

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?

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?

What is the condition for a function f(x) to be differentiable at a point P?

What is the right-hand derivative of f(x) at x = a, denoted by?

What is the set of points where the function f(x) = 1 - e^(-x) is differentiable?

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)?

What is the condition for a function f(x) to be differentiable everywhere?

What is the left-hand derivative of f(x) at x = a, denoted by?

What can be said about the differentiability of the function f(x) = |x-1| at x = 1?

What is the condition for a function f(x) to be differentiable at a point a?

What can be said about the function f(x) if it is differentiable at every point in an open interval (a, b)?

Which of the following functions is everywhere differentiable?

What is the relationship between the left-hand and right-hand derivatives of a function f(x) at a point a?

What is the geometric interpretation of the derivative of a function f(x) at a point a?

What can be said about the function f(x) = 1 + |x| at x = 0?

What is the derivative of the function f(x) = |x - 1| + |x - 3| at x = 2?

What can be said about the function f(x) = |x| at x = 0?

What is the condition for a function f(x) to be differentiable at a point a?

What can be said about the function f(x) = 1 - e^(-x) at x = 0?

What can be said about the function f(x) = |x - 1| + |x + 1|?

What can be said about the function f(x) = (x-1)|x-3|x+2|+cos(|x|) at x = 2?

What is the derivative of f(x) = |x - 1| + |x + 1| at x = 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?

At what points is the function f(x) = |x - 1| + |x + 1| not differentiable?

What can be said about the function f(x) = xe^|x| at x = 0?

What can be said about the function f(x) = 2x - 1 at x = 0?

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?

What is the left-hand derivative of f(x) = |x - 1| + |x + 1| at x = 1?

What can be said about the function f(x) = (x-1)|x-3|x+2|+cos(|x|) for x < 1 or x > 2?

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