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PHY 2048 Learning: Module 03: Differential and Integral Calculus

front 1 Consider the two functions shown below. Assume the blue line represents a(x) and the pink line represents b(x). Which statement is true about these functions at x=3?	back 1
front 2 Find the derivative of U(x) with respect to x for U(x)=4x²-3x+2.	back 2 dU/dx=8x-3
front 3 On the plot shown, assume the horizontal axis represents x and the vertical axis represents y. At what value of x does the function y(x) have a value of zero and a positive derivative?	back 3 The function y(x) is zero at x = 2 and at x = 6 But at x = 2, the function is decresing with negative slope===> negative derivative while at x = 6, the function is incresing with positive slope===> positive derivative
front 4 We know that the velocity (v(t)) is the derivative of position (x(t)) with respect to time, meaning v(t) = dx/dt. Given that, what do we get if we integrate the velocity of an object from t=1 to t=4, meaning integral t=4 t=1 v(t)dt?	back 4 The change in the object’s position between t=1 and t=4

front 5 What is the derivative of B(t) with respect to t for B(t) = Asin(kx-omegat)?	back 5 dB/dt=-omegaAcos(kx-omegat)
front 6 Find the derivative of y(x) with respect to x for y(x)=4+3x.	back 6 dy/dx=3
front 7 The potential energy of a particle experiencing a certain kind of force is given by U(x)=2x+(8/x) measured in joules (J), for positive values of x. What is the minimum potential energy this particle can have?	back 7 8 J
front 8 What is the derivative of U(x) with respect to x for U(x)=10e^5x?	back 8 dU/dx=50e^5x

front 9 What is the correct mathematical translation of the following sentence: “P is the derivative of W with respect to time (t).”?	back 9 P=dW/dt
front 10 What is the rate of change of cos(x) with respect to x?	back 10 −
front 11 On the plot shown, assume the horizontal axis represents x and the vertical axis represents y. What is true about the derivative of the function y(x) at x = 5?	back 11 The derivative of y(x) at x=5 is positive.
front 12 The rate of change of quantity Z is also called its ______.	back 12 derivative Explanation: The rate of change of quantity Z with respect to time is also called its derivative. The integral of a quantity represents summing up the area under the function, not its rate of change. A cross product is a method of multiplying two vectors and is not a rate of change. A dot product is a method of multiplying two vectors and is not a rate of change.

front 13 Which is the correct mathematical representation of the area indicated in the figure?	back 13 The area under a curve is the integral of the function over the indicated range.
front 14 Find the derivative of f(t) with respect to t for f(t)=10t¹⁰.	back 14 Differentiate using the power rule, d/dx[xⁿ]=nx^n-1. df/dt=10 x 10 t ^(10-1) df/dt=100t⁹
front 15 What is d/dx sin(x)?	back 15 cos(x)
front 16 The force exerted on a particle experiencing a certain kind of force is given by F(t)=-3t⁴+8t³+5, with F measured in newtons (N) and t measured in s. At what time is the most force exerted on the particle?	back 16 At about t=2 seconds

front 17 Consider the two functions shown below. Assume the blue line represents a(x) and the pink line represents b(x). Which statement is true about these functions at x=4.5?	back 17
front 18 What is the derivative of E(x) with respect to x for E(x)=Acos(kx)?	back 18 dE/dx=-kAsin(kx)
front 19 On the plot shown, assume the horizontal axis represents x and the vertical axis represents y. At what value of x does the function y(x) have a non-zero value and a zero derivative?	back 19 At x=4
front 20 Find the derivative of a(y) with respect to y for a(y)=2/y² (1+(2y³/5)-(5/y⁴)).	back 20 da/dy=(-4/y³)+(4/5)+(60/y⁷)

front 21 What is the correct mathematical translation of the following sentence: “W is the integral of F with respect to distance (x).”?	back 21
front 22 Consider the two functions shown below. Assume the blue line represents a(x) and the pink line represents b(x). How do the second derivatives of the two functions with respect to x compare?	back 22
front 23 At what value of t will the function v(t)=-3/t² be parallel to the function omega(t)=(3t/4)-5?	back 23 t=2
front 24 Find the derivative of y(x) with respect to x for y(x)=4+3x.	back 24 dy/dx=3

front 25 What is the derivative of v(t) with respect to t for v(t)=3e^t.	back 25 dv/dt=v(t)=3e^t
front 26 Find the derivative of B(t) with respect to t for	back 26 B(t) = 1/t³ - 1/t⁵ + 1/t B'(t) = d (1/t³ - 1/t⁵ + 1/t) / dt = -3/t⁴ + 5/t⁶ -1/t² (d(xⁿ)/dx = n x^(n-1)) n is constant dB/dt = = (-3/t⁴)+ (5/t⁶)-(1/t²)