115pracprob_mid.pdf - MATHEMATICS 115 MIDTERM PRACTICE...

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MATHEMATICS 115MIDTERM PRACTICE PROBLEMS1. Findy0.-1x)lnx
2. Evaluate the following limits.(a)limx0+sinxlnx(b) limx02 sinx-sin 2x2ex-2-2x-x2(c)limx→∞(ex+ 1)1x(d)limx→∞x[ln(2 + 3x)-ln(3x-1)](e)limx→∞cosh 2xe3x-e-3x(f) limx1xx-1-1lnx3.(a) Solve forxgiven that ln(x-3) + ln(x+ 6) = ln 5 + ln 2.(b) Solve fortgiven thaty= 7e(5 ln 2)t.(c) Simplify sin(tan-1xx2+1).(d) Letf(x) =ex+lnx. Show thatf-1exists and find (f-1)0(e) using the formula derivedin class.(e) Find the average value of the functionf(x) =x23-xon [0,2].(f) Find the exact values of (a) cot(tan-1 27), (b) sin-1(sin7π4).(g) Show thatf(x) =ex1-exis 1-1 and find its inversef-1(x).4. Evaluate the integrals.(a)Ze2x+ 1e3xdx(b)Z1202-8x1 + 4x2dx(c)Zdxx(3-lnx)5(d)Zπ/40sec2x2 + tan2xdx(e)Z2dxxp1-4(lnx)2(f)Z5x+ sinxxdx(g)Z41log2xxdx(h)Z5(1 +x2) tan-1xdx(i)Zsinh2(3x)dx5. Find the area of the following region:(a) the region between the curvesy= sinxandy= sin 2x, where 0xπ/2;(b) the region bounded by the curves

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