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In class: d/dx[secx]?
sec x tanx
sec2x
−csc x cot x
tan x
What is the derivative of f(x) = 3x4 - 2x2 + 5?
12x3 - 4x
3x3 - 2x
12x3 - 4x + 5
4x3 - 4x
In a lesson on related rates, a ladder 10 ft long slides down a wall at 2 ft/s. If the base is 6 ft from the wall, how fast is the top sliding?
4/3 ft/s
8/3 ft/s
2 ft/s
3 ft/s
A classroom integral: ∫₀¹ x² dx = ?
1/3
1
0
1/2
Explanation:
To evaluate the definite integral ∫₀¹ x² dx, use the power rule for integration. The antiderivative of x² is (x³)/3. Substitute the upper and lower limits: [(1³)/3] − [(0³)/3] = 1/3 − 0 = 1/3. This represents the exact area under the curve y = x² between x = 0 and x = 1.
Correct Answer:
1/3
Volume by disks: y = x2, rotate about x-axis from 0 to 1?
π/5
π/3
π
2π/5
0
1
∞
Does not exist
Explanation
The real-valued function f(x) = √x is defined only for x ≥ 0. Approaching 0 from the left means considering values x < 0, where f(x) is not defined in the real numbers. Because there are no function values for x < 0, the left-hand limit cannot be evaluated and therefore does not exist as a real limit.
Correct Answer
Does not exist
Implicit: 
Use the first derivative test: f(x) = x4 − 4x2, local min at?
x = ±√2
x = 0
x = ±1
x = ±2
Using the chain rule, find dy/dx if y = sin(3x2).
3x2cos(3x2)
cos(3x2)
6xcos(3x2)
6xsin(3x2)
A teacher demonstrates squeeze theorem: if x2 ≤ sinx/x ≤ 1 near 0, what is limx→0sin x/x?
0
1
∞
-1
Explanation:
By the squeeze theorem, if a function is "sandwiched" between two other functions that have the same limit at a point, then the function also approaches that limit. Here, as x → 0, x2→ 0 and the constant 1 is 1. The standard limit lim x→0 sinx/x is well-known to be 1. Therefore, despite the inequality given, the actual limit is determined by the behavior of sinx/x near 0.
Correct Answer:
1
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