Search
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.08%3A_Math_Review_of_Other_Topics/2.8.15%3A_DerivativesWe start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calcu...We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calculate \(\frac{\Delta f}{\Delta x}\): \[\begin{aligned} \frac{\Delta f}{\Delta x}&=\frac{2x\Delta x+\Delta x^2}{\Delta x}\\ &=2x+\Delta x\end{aligned}\] and take the limit \(\Delta x\to 0\): \[\begin{aligned} \frac{df}{dx}&=\lim_{\Delta x\to 0 }\frac{\Delta f}{\Delta x}\\ &=\lim_{\Delta x\to 0 }(2x+\…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Introductory_Physics_II_(1112)/01%3A_Introduction_to_Physics_and_Measurements/1.02%3A_Thinking_Like_a_ScientistIn a sense, physics can be thought of as the most fundamental of the sciences, as it describes the interactions of the smallest constituents of matter. In practice, the theories of particle physics le...In a sense, physics can be thought of as the most fundamental of the sciences, as it describes the interactions of the smallest constituents of matter. In practice, the theories of particle physics lead to equations that are too difficult to solve for systems that include as many particles as a human brain. One of the key qualities required to be an effective physicist is an ability to understand how to apply a theory and develop a model to describe a phenomenon.
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.10%3A_Anti_derivatives_and_integralsUsing the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0...Using the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0) \Delta x\end{aligned}\] We can then estimate the value of the function \(F_1=F(x_1)\) at the next point, \(x_1=x_0+\Delta x\), as illustrated by the black arrow in Figure A2.3.1 \[\begin{aligned} F_1&=F(x_1)\\ &=F(x+\Delta x) \\ &\approx F_0 + \Delta F_0\\ &\approx F_0+f(x_0)\Delta x\end{aligned}…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/09%3A_Work_Power_and_Energy/9.04%3A_Rolling_motionThe mechanical energy at the bottom of the incline is thus: \[\begin{aligned} E' = K' + U = K'_{rot}+K'_{trans}+(0)=\frac{1}{2}I_{CM}\omega^2 + \frac{1}{2}Mv_{cm}^2\end{aligned}\] Since the disk is ro...The mechanical energy at the bottom of the incline is thus: \[\begin{aligned} E' = K' + U = K'_{rot}+K'_{trans}+(0)=\frac{1}{2}I_{CM}\omega^2 + \frac{1}{2}Mv_{cm}^2\end{aligned}\] Since the disk is rolling without slipping, its angular speed is related to the speed of center of mass: \[\begin{aligned} \omega = \frac{v_{CM}}{R}\end{aligned}\] The moment of inertia of the disk about its center of mass is given by: \[\begin{aligned} I_{CM}=\frac{1}{2}MR^2\end{aligned}\] We can thus write the mecha…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.09%3A_DerivativesWe start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calcu...We start by calculating \(\Delta f\): \[\begin{aligned} \Delta f &= f(x+\Delta x)-f(x)\\ &=(x+\Delta x)^2 - x^2\\ &=x^2+2x\Delta x+\Delta x^2 -x^2\\ &=2x\Delta x+\Delta x^2\end{aligned}\] We now calculate \(\frac{\Delta f}{\Delta x}\): \[\begin{aligned} \frac{\Delta f}{\Delta x}&=\frac{2x\Delta x+\Delta x^2}{\Delta x}\\ &=2x+\Delta x\end{aligned}\] and take the limit \(\Delta x\to 0\): \[\begin{aligned} \frac{df}{dx}&=\lim_{\Delta x\to 0 }\frac{\Delta f}{\Delta x}\\ &=\lim_{\Delta x\to 0 }(2x+\…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_and_Measurements/1.02%3A_Thinking_Like_a_ScientistIn a sense, physics can be thought of as the most fundamental of the sciences, as it describes the interactions of the smallest constituents of matter. In practice, the theories of particle physics le...In a sense, physics can be thought of as the most fundamental of the sciences, as it describes the interactions of the smallest constituents of matter. In practice, the theories of particle physics lead to equations that are too difficult to solve for systems that include as many particles as a human brain. One of the key qualities required to be an effective physicist is an ability to understand how to apply a theory and develop a model to describe a phenomenon.
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.08%3A_Math_Review_of_Other_Topics/2.8.20%3A_Anti_derivatives_and_integralsUsing the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0...Using the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0) \Delta x\end{aligned}\] We can then estimate the value of the function \(F_1=F(x_1)\) at the next point, \(x_1=x_0+\Delta x\), as illustrated by the black arrow in Figure A2.3.1 \[\begin{aligned} F_1&=F(x_1)\\ &=F(x+\Delta x) \\ &\approx F_0 + \Delta F_0\\ &\approx F_0+f(x_0)\Delta x\end{aligned}…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_and_Measurements/1.02%3A_Thinking_Like_a_ScientistIn a sense, physics can be thought of as the most fundamental of the sciences, as it describes the interactions of the smallest constituents of matter. In practice, the theories of particle physics le...In a sense, physics can be thought of as the most fundamental of the sciences, as it describes the interactions of the smallest constituents of matter. In practice, the theories of particle physics lead to equations that are too difficult to solve for systems that include as many particles as a human brain. One of the key qualities required to be an effective physicist is an ability to understand how to apply a theory and develop a model to describe a phenomenon.
