ixed points and stability Slope field Phase line
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- draw the slope field for y' = x² – y², then use it to sketch three solution curves. %|write the function y=2sin(x) as a phase shift of y=2cos(x)Find the tangential component ar and the normal component an of acceleration as a function of t if r(t) = (2r°, 3r³ ) (Express numbers in exact form. Use symbolic notation and fractions where needed.) ат (() an(t) =
- obtain a slope field and add to it graphs of the solution curves passing through the given points. y′ = y^2 with a. (0, 1) b. (0, 2) c. (0, -1) d.(0,0).Determine whether the pairs of the functions are linearly inde-pendent or linearly dependent on the real line. f(x) = e^x sin(2πx), g(x) = e^x cos(2πx)A. Find the derivative of a composite function using the Chain Rule or General Power Rule. 1.) y=(3x+1)³ 2.) y=2(2x+1)⁴ 3.) y= √(3-t) 4.) y= 1/(x-1) 5.) y= x/(√x²-1) 6.) y= ((2x+1)²+1)³ 7.) y=^3√(2x+3) 8.) y= (3x+1)⅔
- Consider the family of curves given by xyey² = c. (a) Use implicit differentiation to express y' in terms of x and y, i.e., write it in the form y' = f(x, y). · y' = y' (b) Write the differential equation for the orthogonal trajectories in the form y' = f(x, y). = у 2xy² + x (c) Find the orthogonal trajectory through the point (1, 1). y = When you solve for y, you'll have to determine whether to use the positive or negative solution. This will be determined by the initial condition.6) use differentials to find approximate valuecalculate f (1,3) with 3 order lagrange
- Determine the speed along a parameterized path c(t) = (6 sin-(t), 3 tan-(t)) at time t = 0 (assume units of meters and seconds). (Use symbolic notation and fractions where needed.) m/s ds dt \1=0Determine whether the given set of functions is linearly independent on the interval (-00, 00). fi (x) = e*, f2(x) = e-*, fg(x) = sinh(x) linearly dependent O linearly independentFind x' for x(t) defined implicitly by x° +t°x+t +4 = 0 and then evaluate x' at the point (- 2,2). .... x' = x'\(-2,2) = (Simplify your answer.)