What is the purpose of Derivatives in Physics and Find the derivative of the functions FG) = (6) FG) = () f) = f(x) =-1 (c) f(x) = f(x) = -2 %3D %3D %3D X-1 1-x
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Q: Find the cross product A x C for (a) A = 2.01 - 4.0j +k and C = 3.0i + 4.0j + 10.0k, (b) A = 3.01 =…
A: 1 )A→ = 2i^ - 4j^ + k^C→ = 3i^ + 4j^ + 10k^
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- w = FjAc = F .Aë , W = m(v;)* - m(u,)* , v 1 zm(us)* - m(v,)* , W = Problem 1: (a) à = 3â – 2ŷ, B = 4â& – 5ŷ. à ·B =? Answer: 22 (b) à = -2â + 4ŷ, B = 3âu – lŷ. Ā·B =? Answer: –10 (c) à = 5â – 3ŷ, B = -3â – 5ŷ. ÷B =? Answer: 0 1 | F,dx , W = F. dř , F, dU W = -AU dx d = 4m F = 20 N 0 = 30° m = 4 kg Hk = 0.5) m = 2 kg F = 40 N Hk = 0.2 Ө — 30° Problem 2: In the figure above on the left we have a force F = 20 N pulling a box to the right at an angle 0 = 30° above the horizontal. The box has a mass m = 2 kg, the coefficient of kinetic firction between the box and the ground is uk = 0.2, and the block moves a distance d = 4 m to the right. (a) What is the work done by the force F? Answer: 69.3 J (b) What is the normal force from the ground? Use g = 10 m/sec² Answer: 10 N (c) What is the work done by the kinetic friction force? Answer: -8 J (d) What is Wnet, the net work done on the box? Answer: 61.3 J (e) The box has an initial speed of 4 m/sec. What is the speed of the box after it has…Ql:a) Find the function? b)Find the derivative of the function by using the derivative definition? ds if s = dt 2t + 1Po = 25 uc/m³ is distributed uniformly through a certain region. If V=50 at y = 2 and V= 90 at y = 5, &, = 2. a- Give the solution and calculate V, E. b- Draw V, Ē, IE with respect to y-axis. c- Now if p₂ = 0, give the solution and calculate V, Ē.
- Problem 1. Evaluate the expression (-2 + 3i)(-3 - 9i) and write the result in the form a + bi. The product isIn Excel, plot R ? as a function of t.Generalizing the Product Rule The Derivative Product Rule gives the formula dv du dr dx (uv) = u- dx for the derivative of the product uv of two differentiable functions of x. a. What is the analogous formula for the derivative of the prod- uct uvw of three differentiable functions of x? b. What is the formula for the derivative of the product uj uz uz u4 of four differentiable functions of x? c. What is the formula for the derivative of a product uj Uz uz* Un of a finite number n of differentiable functions of x?
- Q 1: Let å = -2 ỉ +3 j +6 k and b = 4 i + 6 j , Find ( 2 å + 3 b ,1-ål, å * å , å * b_ )4. Find all second order derivatives for the following functions. (a) f(x,y) = 2.x2 + 3xy + 2y?, (b) f(x,y) =sin (x- 2y).Step 1 Rewrite the original integral Therefore, let du a² + u² (²) dt as + 16 = arctan(y) to Step 2 Use the following integration formula for the inverse trigonometric functions. 1.20 t (27²2² +42 (2) ot = (- t 1 + X X 4 2 (2) dt arctant arctan tan()+c 4 ✓)) + C
- Question 2 Pauli matrices are important in physics. They are: (: :). - 1 0 -i Ox = 1 0 Oy = i 0z = Now consider a vector where 0 is a number and în is a unit vector. Now consider the following matrix: P = ng0x + nyơy + nz0z. Prove that: I2 for k even, P for k odd. 1 Now consider the following matrix exponential: (i0P)™ m! m=0 Show that = I, cos 0 + iP sin 0 . Hint: You may find these two formulas useful: (-1)° 0²p Σ (2p)! (-1)º 0²p+1 sin 0 = Σ cos 0 = (2р + 1)! p=0 p=0For the following compartment model, what is the numerical value of the derivative (in Bq/hr) at t=0, 10, and 100 hours if q(0) = 100 Bq and k = 0.02/hr? Be mathematically precise here. Q--k->1%A9 li. t LTET 4G+ Vol) O A X O U 9:-7 Second H.W.pdf 1. Draw and label the vector with these components. Then determine the magnitude and angle of the vector. a- Az = 3 ,A, = -2 b- B, = -2 ,B, = 2 2. What is the speed of horse in meter per second that runs a distance of 1.2 miles in 2.4 minutes? 3. A car moving along X- axis according on the function: X (t) = 2 t2 + 5t +7 on [1,2] Find: a- The average velocity. b- The acceleration at 3s. 4. Julie is walking around a track at a 2m/s for some exercise. She then decides to start jogging so she accelerates at a rate of 0.5m/s? for 3 seconds. How far did Julie travel from the time she started to accelerate to the end of the 3 seconds? 5. Ralph uses a 2000 to do this. force to push his car along a road for 1000 m. It takes him500 s a- Calculate the amount of work that Ralph did on the car. b- Calculate the amount of power that he generated. II