Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=e^{-x} \sin x$ in $[0, \pi]$ Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=\sin x+\cos x$ in $\left[0, \frac{\pi}{2}\right]$ Solution: Thus, Rolle's theorem is satisfied....
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=\sin 3 x$ in $[0, \pi]$ Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions:. $f(\mathrm{x})=\cos 2 \mathrm{x}$ in $[0, \pi]$ Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=\cos x$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ Solution:...
Read More →A capacitor of capacitance C is given a charge Q.
Question: A capacitor of capacitance $\mathrm{C}$ is given a charge $\mathrm{Q} .$ At $t=0$, it is connected to an ideal battery of emf $\varepsilon$ through a resistance $\mathrm{R}$. Find the charge on capacitor at time $t$. Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(\mathrm{x})=\sqrt{1-\mathrm{x}^{2}}$ in $[-1,1]$ Solution:...
Read More →A capacitor of capacitance C is given a charge Q.
Question: A capacitor of capacitance $\mathrm{C}$ is given a charge Q. At $t=0$, it is connected to an uncharged capacitor of equal capacitance through a resistance R. Find the charge on second capacitor as a function of time. Solution:...
Read More →Consider the situation shown in figure (32-E36).
Question: Consider the situation shown in figure (32-E36). The switch is closed at $t=0$ when the capacitors are uncharged. Find the charge on the capacitor $C_{1}$ as a function of time $t$. Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=(x-2)^{4}(x-3)^{3}$ in $[2,3]$ Solution:...
Read More →A capacitor of capacitance 100μF is connected across
Question: A capacitor of capacitance $100 \mu \mathrm{F}$ is connected across a battery of emf $6.0 \mathrm{~V}$ through a resistance of $20 \mathrm{k} \Omega$ for $4.0 \mathrm{~s}$. The battery is then replaced by a thick wire. What will be the charge on the capacitor $4.0$ s after the battery is disconnected? Solution:...
Read More →The Switch S shown in figure (32-E35) is kept closed for
Question: The Switch S shown in figure (32-E35) is kept closed for a long time and is then opened at $t=0$ Find the current in the middle $10 \Omega$ resistor at $t=1.0 \mathrm{~ms}$ Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=(x-1)(x-2)^{2}$ in $[1,2]$ Solution:...
Read More →Find the charge on each of the capacitors 0.20 ms
Question: Find the charge on each of the capacitors $0.20 \mathrm{~ms}$ after the switch $\mathrm{S}$ is closed in figure (32-E34). Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=(x-1)(x-2)(x-3)$ in $[1,3]$ Solution:...
Read More →A parallel-plate capacitor is filled with a dielectric material having
Question: A parallel-plate capacitor is filled with a dielectric material having resistivity $\rho$ and dielectric constant $\mathrm{K}$. The capacitor is charged and disconnected from the charging source. The capacitor is slowly discharged through the dielectric. Show that the time constant of the discharge is independent of all geometric parameters like the plate area or separation between the plates. Find this time constant. Solution:...
Read More →The value of
Question: By evaluating $\int i^{2} R d t$, show that when a capacitor is charged by connecting it to a battery through a resistor, the energy dissipated as heat equals the energy stored in the capacitor. Solution:...
Read More →A capacitance C charged to a potential difference V is
Question: A capacitance $\mathrm{C}$ charged to a potential difference $\mathrm{V}$ is discharged by connecting its plates through a resistance $\mathrm{R}$. Find the heat dissipated in one time constant after the connections are made. Do this by calculating $\int i^{2} R d t$ and also by finding the decrease in the energy stored in capacitor. Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=x^{3}+3 x^{2}-24 x-80$ in $[-4,5]$ Solution: Thus, Rolle's theorem is satisfied....
Read More →A capacitor of capacitance 12.0μF is connected to
Question: A capacitor of capacitance $12.0 \mu \mathrm{F}$ is connected to a battery of emf $6.00 \mathrm{~V}$ and internal resistance1.00 $\Omega$ through resistance less leads. $12.0 \mu \mathrm{s}$ after the connections are made, what will be (a) The current in the circuit, (b) The power delivered by the battery, (c) The power dissipated in heat and (d) The rate at which the energy stored in the capacitor is increasing. Solution:...
Read More →A capacitor of capacitance C is connected to a battery of
Question: A capacitor of capacitance $C$ is connected to a battery of emf $\varepsilon$ at $t=0$ through a resistance $R$. Find the maximum rate at which energy is stored in the capacitor. When the rate doeshave this maximum value? Solution:...
Read More →How many time constants will elapse before the power delivered
Question: How many time constants will elapse before the power delivered by the battery drops to half of its maximum value in an RC circuit? Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=x^{3}-7 x^{2}+16 x-12$ in $[2,3]$ Solution:...
Read More →How many times constants will elapse before the energy stored in
Question: How many times constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging $\mathrm{RC}$ circuit? Solution:...
Read More →Verify Rolle's theorem for each of the following functions:
Question: Verify Rolle's theorem for each of the following functions: $f(x)=x(x-4)^{2}$ in $[0,4]$ Solution: Condition (1):...
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