Polynomial rational trigonometric logarithmic and exponential functions - One Line Questions
1.
What is the derivative of arccos(x)? —
-1/sqrt(1 - x^2)
2.
What is the limit of (x^2 + 5x + 6) / (x + 2) as x approaches -2? —
-1
3.
Find the derivative of f(x) = cos(3x). —
-3sin(3x)
4.
Find the derivative of f(x) = (x^2 + 1) / (x - 1) using the quotient rule. —
(x^2 - 2x - 1) / (x - 1)^2
5.
What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2? —
4
6.
Evaluate the limit of sin(x)/x as x approaches 0. —
1
7.
What is the limit of (x^3 - 1) / (x - 1) as x approaches 1? —
3
8.
Evaluate the limit of (e^x - 1) / x as x approaches 0. —
1
9.
Evaluate the limit of (1 - cos(x)) / x^2 as x approaches 0. —
1/2
10.
What is the limit of (sqrt(x) - 2) / (x - 4) as x approaches 4? —
1/2
11.
Evaluate the limit of tan(x) / x as x approaches 0. —
1
12.
What is the limit of (e^(2x) - 1) / x as x approaches 0? —
2
13.
Evaluate the limit of (sin(ax)) / (bx) as x approaches 0, where a and b are non-zero constants. —
a/b
14.
What is the limit of (x^2) as x approaches infinity? —
infinity
15.
Evaluate the limit of (ln(x)) as x approaches 1. —
0
16.
What is the limit of (3x^2 + 2x - 1) / (x^2 - 5) as x approaches infinity? —
3
17.
What is the limit of (x - 5) / (x^2 - 25) as x approaches 5? —
1/10
18.
If f(x) = |x|, what is the derivative of f(x) at x = 0? —
Undefined
19.
If f(x) = x * |x|, what is f'(0)? —
0
20.
Find the derivative of f(x) = arctan(x). —
1/(1 + x^2)
21.
What is the derivative of arcsin(x)? —
1/sqrt(1 - x^2)
22.
If f(x) = log_b(x), what is its derivative? —
1/(x*ln(b))
23.
What is the derivative of f(x) = log_a(x^2)? —
2/(x*ln(a))
24.
Find the derivative of f(x) = 5x^4 - 3x^2 + 2x - 7. —
20x^3 - 6x + 2
25.
Find the second derivative of f(x) = x^3. —
6x
26.
What is the limit of (x^n - a^n) / (x - a) as x approaches a? —
na^(n-1)
27.
If a function is differentiable at a point, it must be: —
Continuous
28.
The function f(x) = |x - 2| is: —
Continuous but not differentiable at x = 2.
29.
The function f(x) = x^3 is: —
Continuous and differentiable everywhere.
30.
The function f(x) = 1/x is: —
Discontinuous at x = 0.
31.
The function f(x) = x^3 - x is: —
Continuous and differentiable for all real numbers.
32.
Find the derivative of f(x) = sin(x^2). —
2x*cos(x^2)
33.
If lim (x->a) f(x) = L and f(a) = L, then f(x) is: —
Continuous at x = a.
34.
Find the derivative of f(x) = e^(sin(x)). —
cos(x) * e^(sin(x))
35.
If f(x) = e^x, what is the derivative of f(x)? —
e^x
36.
For a function f(x) to be continuous at x = c, which of the following conditions must be met? —
All of the above.
37.
Which of the following functions is NOT continuous at x = 0? —
f(x) = 1/x
38.
Which condition is NOT required for a function f(x) to be differentiable at x = c? —
The derivative from the left must equal the derivative from the right.
39.
A function f(x) is differentiable at x = c if: —
All of the above.
40.
Which of the following is NOT a property of limits? —
lim (x->c) f(g(x)) = f(lim (x->c) g(x)) if f is continuous at lim (x->c) g(x)
41.
What is the derivative of the function f(x) = x^n? —
nx^(n-1)
42.
What is the derivative of tan(x)? —
sec^2(x)
43.
What is the derivative of ln(x) with respect to x? —
1/x
44.
What is the derivative of f(x) = sqrt(x^2 + 1)? —
x / sqrt(x^2 + 1)
45.
Consider the function f(x) = x^2 if x is rational and f(x) = 0 if x is irrational. At which point is f(x) continuous? —
x = 0
46.
What is the derivative of f(x) = x^x? —
x^x * (1 + ln(x))
47.
If f(x) = a^x, where a > 0 and a != 1, what is its derivative? —
a^x * ln(a)
48.
Consider the function f(x) = x^2 for x < 0 and f(x) = x for x >= 0. Is f(x) continuous at x = 0? —
Yes, because f(0) = 0 and the limits from both sides are 0.
49.
Determine if the function f(x) = x^2 is continuous at x = 3. —
Yes, because f(3) = 9 and lim (x->3) f(x) = 9.
50.
Consider the function f(x) = x^2 sin(1/x) for x != 0 and f(0) = 0. Is f(x) differentiable at x = 0? —
Yes, the derivative is 0.