Question Bank: Continuity and Differentiability

What is the definition of continuity at a point c for a function f?

Which of the following functions is continuous at x = 2?

At which point is the function f(x) = {2 if x < 1, 3 if x ≥ 1} discontinuous?

Which condition is necessary for differentiability at a point?

How do you identify a removable discontinuity in a function?

Which of the following functions is not continuous?

If f is continuous at c, which statement is always true?

Which of the following statements is false regarding continuity?

What does the Intermediate Value Theorem state?

A function can be differentiable at a point but not continuous at that point. True or False?

What is an example of a step function?

If a function f has a removable discontinuity at x = c, which of the following is true?

Which of the following characterizes a non-removable discontinuity?

Which theorem can help analyze the continuity of functions?

Which of the following is the domain of the exponential function f(x) = b^x, where b > 0?

What is the range of the exponential function f(x) = b^x, where b > 1?

Which of the following functions is continuous for all real numbers?

The value of b in the function f(x) = b^x must be:

To determine the continuity of f(x) = 3x - 4 at x = 2, what should you evaluate?

If f(x) = 2^x, what is the value of f(0)?

The function f(x) = x^2 - 3 is continuous because it is which type of function?

What can be inferred about the growth rate of f(x) = 10^x compared to polynomial functions?

At which point is the function f(x) = (x^2 - 1)/(x - 1) discontinuous?

What is the derivative of the function f(x) = e^x?

For f(x) = |x - 3|, what can be said about its continuity at x = 3?

Which of the following is the logarithmic form of the equation 2^3 = 8?

If f(x) = 2x + 5, what type of function is it regarding continuity?

If f(x) = x^2 and g(x) = 2^x, what is the growth comparison of f and g for large x?

Examine the function f(x) = 1/(x^2 - 4). Is it continuous?

What is the value of ln(e^5)?

For f(x) defined as f(x) = x + 1 if x ≤ 1 and f(x) = 3 if x > 1, is f continuous at x = 1?

What is the base of the natural logarithm?

Which statement correctly describes a continuous function?

Which function demonstrates logarithmic growth as opposed to exponential growth?

Which function is a classic example of a discontinuous function?

The inverse of the exponential function f(x) = b^x is:

The function f(x) = sqrt(x-1) is continuous at which of the following points?

What happens to f(x) = 1/b^x as x approaches infinity, where b > 1?

To prove that f(x) = x^3 - 2x^2 + x is continuous at x = 2, you must show what?

Which equation represents the relationship between exponential and logarithmic functions?

A function that is continuous everywhere is often termed what?

Consider the function f(x) = |x|. Is it continuous?

What is the derivative of the function f(x) = 3x^2?

Which of the following statements defines differentiability at a point c?

If f(x) = |x|, what is f'(0)?

Which of the following functions is not differentiable at x = 0?

What is the derivative of f(x) = sin(x) at x = 0?

For the function f(x) = x^2, what is the value of f’(2)?

Which of the following functions is guaranteed to be differentiable on its entire domain?

What is the second derivative of f(x) = x^3 - 3x^2?

Why is the function f(x) = ln(x) not differentiable at x = 0?

If f(x) is differentiable at x = c, which of the following must also be true?

What does the Mean Value Theorem state?

What is the main reason that a polynomial function is differentiable everywhere?

Which function is differentiable everywhere except at one point?

At which point is f(x) = x^2 - 4x + 4 not differentiable?

What value does f'(x) = 3x^2 equal at x = 1?

What is the first step in logarithmic differentiation?

If y = x^x, what is the derivative dy/dx using logarithmic differentiation?

What condition must y and u(x) satisfy for logarithmic differentiation to be applicable?

Given y = (sin x)^x, compute dy/dx using logarithmic differentiation.

What is the final expression for the derivative of y = e^(x^2) using logarithmic differentiation?

If y = (x^2 + 1)^(sin x), what is dy/dx?

Using logarithmic differentiation, what is the derivative of y = x^x?

If y = (2x + 3)^(x^2), what is dy/dx using logarithmic differentiation?

When is logarithmic differentiation particularly useful?

For the function y = (x + 1)/(x - 1), what is dy/dx using logarithmic differentiation?

Which of the following equations represents the correct use of logarithmic differentiation?

What is the derivative of y = ln(x^x)?

If y = (x^3 + 2)^(1/x), what is dy/dx using logarithmic differentiation?

If x = t^2 and y = 3t, what is dy/dx?

For x = 2cos(t) and y = 2sin(t), what is the value of dy/dx?

If x = e^t and y = e^(2t), how would you express dy/dx?

For the parametric equations x = t^3 and y = t^2 - t, find dy/dx.

If x = sin(t) and y = cos(t), what is dy/dx?

Given x = a(t + sin(t)) and y = a(1 - cos(t)), find dy/dx.

For the parametric equations x = ln(t) and y = t^2 + t, find dy/dx.

If x = 3t and y = 4t^2, calculate dy/dx.

For x = e^(-t) and y = sin(t), find dy/dx.

Given x = t^5 and y = t^3 - t, compute dy/dx.

If x = t^2 - 4t and y = t^3 - 3t, what is dy/dx?

For x = sin(t) + cos(t) and y = sin(t) - cos(t), find dy/dx.

If x = t^3 and y = ln(t), calculate dy/dx.

For x = tan(t) and y = sec(t), find dy/dx.

If x = 2t^2 and y = t^3 + 1, what is dy/dx?

For x = t^4 and y = t^2, calculate dy/dx.

What is the second order derivative of y = x^3 + tan x?

If y = 3e^(2x) + 2e^(3x), what is the second order derivative d^2y/dx^2?

Show that for y = A sin x + B cos x, d^2y/dx^2 + y = 0.

For the function y = sin^(-1)(x), what is d^2y/dx^2?

Find d^2y/dx^2 if y = x^4.

If y = e^(5x), what is d^2y/dx^2?

Prove that d^2y/dx^2 + y = 0 for y = 5 cos x - 3 sin x.

What is the second derivative of y = x^2 log x?

If y = cos^(-1)(x), find d^2y/dx^2.

Prove that d^2y/dx^2 = 0 for y = A e^(mx) + B e^(nx) if m = n.

What is the second derivative of y = x^2 sin x?

Find d^2y/dx^2 for y = tan^(-1)(x).

If y = ln(x), then what is d^2y/dx^2?

Show that for y = 3cos(log x) + 4sin(log x), d^2y/dx^2 + xy = 0.