Math 1A - Fall 2005 - Borcherds - Midterm 1

# Math 1A - Fall 2005 - Borcherds - Midterm 1 - 8832832885...

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Unformatted text preview: 8832832885 15:83 5188422415 LICE MnﬂiIN LIERnﬂiR‘r’ FWSE Math 1A Midterm 1 2005-9-29 11:00-12:3'opm. R.eeroherd5 You are allowed 1 sheet of notes. Calculators are not allowed. Each question is worth 3 marks, which will only be given for a clear and correct answer. 1. 2. 10. 11. 12. 13. 14-1. 15. Find the domain of the function 9(a) = we —|— ME — u. Sketch the graph of y = Id:2 — Ecol. Find a formula. for the inverse of the function f (m) = 1 —|- em3 Sketch the graph of a function f that satisﬁes the conditions lirn ﬂea )= 1,11n1 ﬁx) = —1,Iirn ffa‘) = 1,11111 ﬁe): m1,f(2) = 1. 2—98“ gaff—40+ sou-1— sc—v-l 1+ ,1 Evaluate the limit 3:3 — 1 lim 3—1-1 332 —1 How close to 2 do wehave to take a: so that 5:1: + 3 is within a distance of 0.01 from 13? Find the numbers at which f is discontinuous, where f is deﬁned by f(:1:) = m2 if 33:5 1, f(m) =1/znif1 {cc-:13, f(m) =1/2+w’:n—3if:c23. Whatis 3 1 4 1 lim (:13+)(:r+) {CHM-DO W A curve has equation 1 = f (1:). Write an expreSsion for the slope of the secant line through the points (3, f(3)) and (FE, f (113)), and write an expression for the slope of the tangent line at (3, f(3)). If 9(5):) = 51:3 + m2 + a: + 1, ﬁnd g’m) and use it to ﬁnd an equation of the tangent line to the curve 1; = m3 + 11:2 + :c + 1 at the point (D, 1). Sketch the graph of the function y = x2|m|, say where it is differentiable, and ﬁnd a formula for its derivative. Differentiate the function y = e1“H + 4W2 + (a: + 1) / v5 At What point on the curve y = l + 26" — 3:1: is the tangent line parallel to the line 39’: — y = 5? 1 Differentiate areI (1/? + 1) Diiferentiate 1+1/3: 81.-"81 ...
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