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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 55b
Chapter 3, Problem 55b

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>
b. Is Q′ positive or negative for t≥0?

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Step 1: Understand the problem context. The problem involves a capacitor in an electrical circuit where the charge Q on the capacitor varies over time. We need to determine the sign of the derivative of Q, denoted as Q', for t ≥ 0.
Step 2: Recall that the derivative Q' represents the rate of change of the charge Q with respect to time t. If Q' is positive, it means the charge is increasing over time. If Q' is negative, it means the charge is decreasing over time.
Step 3: Analyze the behavior of the charge Q over time from the given figure (not provided here). Look for whether the charge is increasing or decreasing as time progresses for t ≥ 0.
Step 4: If the graph of Q versus t shows an upward trend for t ≥ 0, then Q' is positive. Conversely, if the graph shows a downward trend, then Q' is negative.
Step 5: Conclude based on the observed trend in the graph. If the graph is flat (neither increasing nor decreasing), then Q' would be zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage. It is defined as C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the capacitor. Understanding capacitance is essential for analyzing how charge varies with time in a circuit.

Charge Variation

The variation of charge (Q) over time in a capacitor is influenced by the current flowing into or out of it. The rate of change of charge, denoted as Q′, indicates whether the charge is increasing or decreasing. A positive Q′ means the charge is increasing, while a negative Q′ indicates a decrease.
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Current and Charge Relationship

The relationship between current (I) and charge (Q) is given by the equation I = dQ/dt, where dQ/dt represents the rate of change of charge with respect to time. This relationship is crucial for determining whether Q′ is positive or negative, as it directly links the flow of current to the change in charge on the capacitor.
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Related Practice
Textbook Question

Robert Boyle (1627–1691) found that for a given quantity of gas at a constant temperature, the pressure P (in kPa) and volume V of the gas (in m³) are accurately approximated by the equation V=k/P, where k>0 is constant. Suppose the volume of an expanding gas is increasing at a rate of 0.15 m³/min when the volume V=0.5 m³ and the pressure is P=50 kPa. At what rate is pressure changing at this moment?

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Textbook Question

Evaluate and simplify y'.

x = cos (x−y)

Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

At what times and positions is the velocity zero?

Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.

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Textbook Question

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

a. At what time is the rate of change of the charge Q' the greatest?

Textbook Question

Calculate the derivative of the following functions.

y = cos7/4(4x3)