Concept explainers
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CALCULUS+ITS APPLICATIONS
- Let V = span{e2", xe2ª , x²e2¤}. (a) Show that d dx + agze?r + аҙӕ*е2) € V for any aj, az, aҙ € R, (b) Let 0 and represent the functions e2a, xe2* and x²e2x. respectively. For example, 4 represents 3e2 + 4xe2a + 5x²e2ª. 5 Find the matrix of differentiation as a linear transformation on V.arrow_forwardLet u = (6,0, -2) and Vf(a) = (-2,0,-8). What is the value of Du[f] (a)? Du f(a) = Numberarrow_forwardFind Vf(a), given the following data. Calculate 91, 92 and g3 such that Vƒ(a): 91 92 93 = || || = 91 92 93 2 u1 »-(-:-) = u2= uz = = -5 and Du₁ [f] (a) = −5 1 -3 -3 1 -5 and Dug [f] (a) = 20 and Dug [f] (a) = 2arrow_forward
- "Hand-in question": While biking, the force of air resistance F (mass times length, per time- squared) appears to be positively related to the speed v (length per time) at which the bike travels and the cross-sectional area A (length-squared) of bike and biker. However, the force involves mass and hence cannot depend on v and A alone. Let the force also depend on the air density p (mass per length-cubed). Hence, the force F can be expressed as some multiplicative combination of v, A and p, that is F = kv" Aª p², for k E R. Find the values of r, y and z that result in the above equation being dimensionally consistent.arrow_forwardWhat does the gradient vector of a function represent? Create your own real-life example that demonstrates its potential utility.arrow_forward
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