Scaling (Quadratic Forms) There are instances in Physics 1 where the functional form of an expression...

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Scaling (Quadratic Forms) There are instances in Physics 1 where the functional form of an expression goes as a parameter squared. One example you will encounter in Physics 1A is that of the potential energy of a spring system which goes as PE = k x, where x is the displacement from equilibrium. In this exercise, we will explore the consequences of having a computed value that is proportional to the square of a given or measured parameter. 15. Let's consider the example of the potential energy of the spring. (a) In the table provided, in column 2 calculate the PE for the values given in the first column for k = 1. X -6 654 PE 1/2 kx2 APE = 18 12.5 -5 -4 -3 4.5 -2 2 -1 0 1 2 3 4.5 4 5 0.5 6 (b) In the graph below, plot the PE data over the displacement range provided. Determine the best values for the domain (x values) and range (y values) before plotting. Sketch out the functions given the values in the table. -2 4 11 At what pair of x values are the PE the same?_ (can you generalize?) (c) For the calculation of the PE, if you double the value for x, how much larger is the value for the new PE' (ie. what is PE'/PE)? (d) If you double the value for k, how much larger is the value for the new PE' (ie. what is PE'/PE)? (e) Compare the difference in the PE between consecutive values of x and plot in column 3 of the table. How does the PE difference between x = 2 and x = 1 compare to the difference between x = 1 and x = 0? (f) How does the PE difference between x = 3 and x = 2 compare to the difference between x = 2 and x = 1? 14. For the function f shown in the upper graph, generate the integral ((fdx) in the second graph below: 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 xpfj 69 X 88 X 10 12 Calculate the derivative of the following functions: f=5x, df/dx S f=5x, df/dx = 10X f=5x+4, df/dx 10% Calculate the integral (indefinite) of the following functions: f=5x, fdx= =5x, [fdx= 3x/2 5x313 f=5x+4, ffdx = 5x3 + 4x 10 Calculus Here we review basic calculus that will be important for determining, for example, velocity as displacement over time, work as a function of which is force over distance, etc. Key concepts: 13. . Derivative measures the slope of a function Integration measures the accumulation of a function (area under a curve) Important functions (A, k, and c are constants): o polynomials: d/dx Ax" = nAx"-, SAx" dx = (A/n+1)x+1 + c (for n-1) o sine: d/dx Asin(kx) = Akcos(kx), Asin(kx) dx = -(A/k)cos(kx) + c o cosine: d/dx Acos(kx) = -Aksin(kx), Acos(kx) dx = (A/k)sin(kx) + c o exponential: d/dx Aekx = Akekx, SAekx dx = (A/k)ekx + c o logarithm: d/dx Aln(x) = Ak/x, A/x dx = Aln(x) + c For the function f shown in the upper graph, generate the derivative (df/dx) in the second graph below: 1.4 1.2 df/dx 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 X 10 10 12 9 Scaling (Quadratic Forms) There are instances in Physics 1 where the functional form of an expression goes as a parameter squared. One example you will encounter in Physics 1A is that of the potential energy of a spring system which goes as PE = k x, where x is the displacement from equilibrium. In this exercise, we will explore the consequences of having a computed value that is proportional to the square of a given or measured parameter. 15. Let's consider the example of the potential energy of the spring. (a) In the table provided, in column 2 calculate the PE for the values given in the first column for k = 1. X -6 654 PE 1/2 kx2 APE = 18 12.5 -5 -4 -3 4.5 -2 2 -1 0 1 2 3 4.5 4 5 0.5 6 (b) In the graph below, plot the PE data over the displacement range provided. Determine the best values for the domain (x values) and range (y values) before plotting. Sketch out the functions given the values in the table. -2 4 11 At what pair of x values are the PE the same?_ (can you generalize?) (c) For the calculation of the PE, if you double the value for x, how much larger is the value for the new PE' (ie. what is PE'/PE)? (d) If you double the value for k, how much larger is the value for the new PE' (ie. what is PE'/PE)? (e) Compare the difference in the PE between consecutive values of x and plot in column 3 of the table. How does the PE difference between x = 2 and x = 1 compare to the difference between x = 1 and x = 0? (f) How does the PE difference between x = 3 and x = 2 compare to the difference between x = 2 and x = 1? 14. For the function f shown in the upper graph, generate the integral ((fdx) in the second graph below: 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 xpfj 69 X 88 X 10 12 Calculate the derivative of the following functions: f=5x, df/dx S f=5x, df/dx = 10X f=5x+4, df/dx 10% Calculate the integral (indefinite) of the following functions: f=5x, fdx= =5x, [fdx= 3x/2 5x313 f=5x+4, ffdx = 5x3 + 4x 10 Calculus Here we review basic calculus that will be important for determining, for example, velocity as displacement over time, work as a function of which is force over distance, etc. Key concepts: 13. . Derivative measures the slope of a function Integration measures the accumulation of a function (area under a curve) Important functions (A, k, and c are constants): o polynomials: d/dx Ax" = nAx"-, SAx" dx = (A/n+1)x+1 + c (for n-1) o sine: d/dx Asin(kx) = Akcos(kx), Asin(kx) dx = -(A/k)cos(kx) + c o cosine: d/dx Acos(kx) = -Aksin(kx), Acos(kx) dx = (A/k)sin(kx) + c o exponential: d/dx Aekx = Akekx, SAekx dx = (A/k)ekx + c o logarithm: d/dx Aln(x) = Ak/x, A/x dx = Aln(x) + c For the function f shown in the upper graph, generate the derivative (df/dx) in the second graph below: 1.4 1.2 df/dx 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 X 10 10 12 9