Problem Sets – Differentiation and Linear ApproximationThese problem sets contain additional practice problems that correspond to the lecture material. There are hundredsmore in the calculus textbooks on reserve at the library, or on the Davis shelves close to QA303.Differentiation1. For each of the following, determinedydx.a)y=x2e√xb)y= ln(cos(x3))c)y=pcos3(x2) + 7d)y= (ln (sinx))√xe)exy2-ysin(x3y)= 12. For each of the following, determinedydx, assuming thatf(x) andg(x) are positive, differentiable functions.a)y= ln(f(x2))b)y= [f(x)]g(x)c)y=f(√xln (g(x)))3. Ify= (arcsin(x))2and 0< x <1, then prove that(1-x2)dydx2= 4y,and thereby deduce that(1-x2)d2ydx2-xdydx-2 = 0.Hint:To deduce the second expression, try differentiating the first expression with respect tox.4. a) Approximate the cosine function cosxby the polynomiala0+a1x+a2x2, where the three coefficients aredetermined from the conditions,dndxncosxx=0=dndxn(a0+a1x+a2x2)x=0forn= 0,1,2.b) Calculate cos 15◦using this polynomial.c) How would you generalize this procedure to higher degree polynomials,anxn, (N >2)?NXn=0Linear ApproximationNewton’s Method5. Find the root of sinx= 1-xto 5 decimal places. Use a sketch to estimate the initial pointx.6. How many roots does the equation tanx=xhave? Find the one betweenπ2and3π2to 5 decimal places.Limits7. Evaluate the following indeterminate forms using l’Hˆopital’s rule.a)limx→πxsinxx-πb)limx→π2ln (2-sinx)ln (1 + cosx)c)limx→∞lnx√xd) limx→xεlnx(ε >0)e)limx→∞xre-x(r∈R, r >0)8. What is wrong with the following use of l’Hˆopital’s rule:limx→1=x3+x-2x2-3x+ 2= limx→13x2+ 12x-3= limx→16x2= 3(The limit is actually-4).