MATH 2100 C958 Calculus I
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Free MATH 2100 C958 Calculus I Questions
Evaluate limₓ→0 (eˣ − 1)/x using known limits.
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0
-
e
-
1
-
∞
Explanation
Explanation:
The limit limₓ→0 (eˣ − 1)/x is a standard derivative-based limit derived from the definition of the derivative of eˣ at x = 0. Since the derivative of eˣ is eˣ, substituting x = 0 gives e⁰ = 1. Therefore, this limit equals 1.
Correct Answer:
1
Implicit: xy = 1, find dy/dx.
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−x/y
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y/x
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x/y
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−y/x
Explanation
Implicit: x3 + y3= 3xy, dy/dx at (1,1)?
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1
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-1
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0
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Undefined
Explanation
In teaching related rates, a teacher emphasizes that all rates must be:
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With respect to the same variable (usually time)
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Constant
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Positive
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In terms of x only
Explanation
Explanation:
In related rates problems, all rates of change must be expressed with respect to the same independent variable, usually time (t), to ensure that the relationships between the changing quantities are consistent. This allows differentiation of equations that relate the variables and ensures that the computed rates are meaningful and comparable. Without using the same variable, the chain of differentiation would be incorrect and lead to invalid results.
Correct Answer:
With respect to the same variable (usually time)
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0
-
1
-
3
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∞
Explanation
Critical points: f(x) = lnx + x2, domain x > 0.
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x = 1/√2
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x = 1
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x = 1/2
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x = √2
Explanation
Arc length of y = x3/2 from x = 0 to 1?
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13/27 (√3 + 1)
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8/27 (√3 + 1)
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13/27 (3√3 - 1)
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1
Explanation
Related rates: water poured into an inverted cone r = h/3 at 6 ft³/min, dh/dt when h = 3?
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2/π ft/min
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1/π ft/min
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6/π ft/min
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3/π ft/min
Explanation
In a classroom, the derivative of f(x) = ln(x² + 1)?
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2x / (x² + 1)
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1 / (x² + 1)
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x / (x² + 1)
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2 ln(x² + 1)
Explanation
Explanation:
To find the derivative of f(x) = ln(x² + 1), we apply the chain rule. The derivative of ln(u) is 1/u × du/dx. Here, u = x² + 1, so du/dx = 2x. Therefore, f′(x) = (1 / (x² + 1)) × 2x = 2x / (x² + 1).
Correct Answer:
2x / (x² + 1)
Linear approximation for f(x) = √x at x = 4: approximate √4.1.
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2.025
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2
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2.05
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2.1
Explanation
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