Limits
A function has a limit as x --> c. What must be true?
The left-hand and right-hand limits must be equal.
Let f(x)=∣x∣
Which statement best explains why f is not differentiable at x=0?
A) f is not continuous at x=0.
B) The left-hand and right-hand derivatives at x=0 are different.
C) f(0) is undefined.
D) The derivative exists and equals 0.
B) The left-hand and right-hand derivatives at x=0 are different.
Which set of conditions guarantees that a value k between f(a) and f(b) is achieved by some c ∈ [a,b]?
A) f is differentiable on [a,b], and k lies between f(a) and f(b)
B) f is continuous on [a,b], and k lies between f(a) and f(b)
C) f is continous on (a,b), and f(a) and f(b) have opposite signs
D) f is differentiable on (a,b), and k lies between f(a) and f(b)
B) f is continuous on [a,b], and k lies between f(a) and f(b)
If f(x) = sin(x)
What is f'(x)?
A) sin(x)
B) -sin(x)
C) cos(x)
D) -cos(x)
C) cos(x)
A function is continuous at x=c. Name the three conditions that must be satisfied.
If f'(c)=0, does that guarantee a local maximum or minimum at x=c?
No. It could be a local max, local min, or neither (like a flat point or inflection point).
Let f(x) = x^3 - 2x^2 on [0,2]. Which value of c ∈ (0,2) is guaranteed by the mean value theorem?
A) c = 4/3
B) c = 0
C) c = 2
D) No such c exists
A) c = 4/3
if y= arcsin(2x)
Then dy/dx is:
A) 2/√(1-4x^2)
B) 1/√(1-4x^2)
C) 2/1-4x^2)
D) 1/1-4x^2
A) 2/√(1-4x^2)
Let f(x) = (x² − 1)/(x − 1) for x ≠ 1, and f(1) = 3. Classify the discontinuity at x = 1. If removable, what value of f(1) fixes it?
A) Removable; f(1) = 2 makes f continuous
B) Removable; f(1) = 1 makes f continuous
C) Jump discontinuity; no value of f(1) can fix it
D) Infinite discontinuity, due to the factor (x − 1) in the denominator
A) Removable; f(1) = 2 makes f continuous
A particle moves along a line with velocity
v(t) = t^2 -4t + 3
for t≥0
on which interval(s) is the particle slowing down?
A) (0,1) ∪ (2,3)
B) (1,2) ∪ (3,∞)
C) (0,2)
D) (1,3)
A) (0,1) ∪ (2,3)
A function is continuous on [a,b], differentiable on (a,b), and f(a)=f(b). What does Rolle’s Theorem guarantee, and what is the strongest possible conclusion about the graph?
At least one c in(a,b) where f'(c)=0; the graph must have a flat tangent somewhere inside the interval.
Let f(x) = sin^2 (x) + cos^2 (x). Find f'(x)
A) 0
B) 1
C) 2 sinx cosx
D) -2 sinx cosx
A) 0
Evaluate lim(x→∞) (3x³ − 2x + 1)/√(9x⁶ + 5).
A) 1/3
B) 3
C) 0
D) 1
D
Let g(x) = ∫ from 1 to x |t-2| dt
Which statement is true?
A) g is not continuous at x = 2
B) g is continuous but not differentiable at x =2
C) g is differentiable at x=2
D) g'(2) does not exist
C) g is differentiable at x=2
Let f be differentiable everywhere, with f(0) = 2, f(2) = 2, and f(4) = 8
Which conclusion must be true?
A) there exist distinct points c1 ∈ (0,2) and c2 ∈ (2,4) such that f'(c1) = 0 and f'(c2) = 3
B) f'(x) must be constant on (0,4)
C) there exists a single point where f'(x) = 0 and f'(x) = 3
D) f is not differentiable at some point in (0,4)
A) there exist distinct points c1 ∈ (0,2) and c2 ∈ (2,4) such that f'(c1) = 0 and f'(c2) = 3
If f(x) = e^tan^2x, Then f'(x) =
A) e^tan^2 (x)
B) sec^2 (x) e^tan^2 (x)
C) tan^2 (x) e^tan^2 (x-1)
D) 2 tan (x) sec^2 (x) e^tan^2 (x)
D) 2 tan (x) sec^2 (x) e^tan^2 (x)