# Final math practice.pdf - Question 1 Z)x p Let f(x = 1 t2...

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Question 1.()Letf(x) =Zx1p1 +t2dt, x2R.(a) Compute the derivativef0(x).(b) Show thatf(x) has an inverse functionf-1(x) by showingf(x) is one-to-one.(c) Findf-1(0).(d) Compute the derivative (f-1)0(0). You may use the formula (f-1)0(a) =1f0(f-1(a)).Page 2
Question 2.()Give a definition of Euler’s constant, that is the numbere= 2.718. . .Question 3.()Using logarithmic dierentiation to find the derivative ofy=2xcosh(x)px2+ 1Question 4.()Consider the conic section 4x2= 64-16y2.(a) Locate thexandy–axes intercepts.(b) Identify whether the curve is an ellipse or a hyperbola, and sketch it.(c) Determine the vertices and foci.Page 3
Question 5.()Evaluate the following limits using appropriate methods.(a)limn!1(p2n+ 3-p2n+ 1)(b) limx!0(ln(x))2x(c) limx!0sin(x)-x+x36x5Page 4
Question 6.()Evaluate the following integrals.(a)Zsin3(x) cos4(x)dx(b)Zt3et2dtPage 5
(c)Zx-4x2-5x+ 6dx(d)Zxp3-2x-x2dxPage 6