137_PS4_Derivatives.pdf - Problem Sets \u2013 Differentiation and Linear Approximation These problem sets contain additional practice problems that

# 137_PS4_Derivatives.pdf - Problem Sets u2013...

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Problem Sets – Differentiation and Linear Approximation These problem sets contain additional practice problems that correspond to the lecture material. There are hundreds more in the calculus textbooks on reserve at the library, or on the Davis shelves close to QA303. Differentiation 1. For each of the following, determinedydx.a)y=x2exb)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 15using this polynomial. c) How would you generalize this procedure to higher degree polynomials,anxn, (N >2)? NXn=0 Linear Approximation Newton’s Method 5. 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. Limits 7. Evaluate the following indeterminate forms using l’Hˆopital’s rule.a)limxπxsinxx-πb)limxπ2ln (2-sinx)ln (1 + cosx)c)limx→∞lnxxd) limxxεlnx(ε >0)e)limx→∞xre-x(rR, r >0)8. What is wrong with the following use of l’Hˆ opital’s rule:limx1=x3+x-2x2-3x+ 2= limx13x2+ 12x-3= limx16x2= 3(The limit is actually-4).