MAS161: Calculus & Matrix Algebra Semester 2, 2015 ASSIGNMENT 1 Due by 4:00pm, Friday 21 August Total Marks: 65 1. [9 marks] Differentiate the following functions. (a) f ( x)= 3x 1 2 e 2x-1 (b) g( x)= e x 2 cos( x 2 ) (c) y( x)= 2x - 1 2x 3 - 1 2. [6 marks] Find the maximum possible area of a rectangle that has one side along the x-axis and its upper vertices on the function y = 27 - 3x 2 . Include a sketch. 3. [8 marks] Use implicit differentiation to find the equation of the tangent line to the curve 1 3 xy 2 + p 6xy = 10 at (3, 2). 4. [10 marks] (a) Without doing a sketch, show that the cubic equation x 3 + x 2 + x - 1 = 0 has at least one solution on the interval [0, 1]. [Hint: Use a theorem discussed in lectures, or see Section 1.8 of Calculus (7th ed) by Stewart.] (b) Now, by sketching the cubic x 3 + x 2 + x - 1 (by hand or by computer), you should see that there is, in fact, exactly one zero in the interval [0, 1]. Use Newton’s method to find this zero accurate to 3 decimal places. You should include a sketch of the cubic, Newton’s iteration formula, and the list of iterates. [Use a computer if possible, e.g., a spreadsheet or MATLAB.] 5. [5 marks] Using L’Hˆ opital’s rule (or otherwise), determine the following limits. (a) lim x→0 1 - cos 3x 2 2x 2 (b) lim x→1 ln(4 - 3x) x - 1 6. [23 marks] Evaluate the following integrals, showing all your working. (a) R x ln(1 + 2x 2 ) dx (b) R 3 √ 16-4x 2 dx (c) R 6 4 2s (s-1)(s+2) ds (d) R 2 1 x x 2 +2x+5 dx (e) R ∞ 1 2x 2 e -x dx [Hint: For the improper integral in part (e), you can use the fact that exponentials grow faster than polynomials; in particular, lim x→∞ p( x)/e x = 0 where p( x) is a polynomial function of x.] 7. [4 marks] Compute the following integral, accurate to 3 decimal places, using any technique that you think is appropriate. Show all your working. Z 1 2 0 e -x 2 dx ???????
MAS161: Calculus & Matrix AlgebraSemester 2, 2015ASSIGNMENT
1Due by 4:00pm, Friday 21 August Total Marks: 651. [9 marks]
Differentiate the following functions.(a) f (x) = 3x12e2x1(b) g(x)
= ex2 cos(x2)(c) y(x) =2x 12x312. [6 marks] Find the maximum
possible area of a rectangle that has one side along the x-axis
andits upper vertices on the function y = 27 3x2. Include a
sketch.3. [8 marks] Use implicit differentiationtondthe equationof
the tangent line tothe curve13xy2+
6xy = 10 at (3, 2).4. [10 marks](a) Without doing a sketch, show
that the cubic equation x3+ x2+ x 1=0 has at least onesolution on
the interval [0, 1].[Hint: Use a theorem discussed in lectures, or
see Section 1.8 of Calculus (7th ed) by Stewart.](b) Now, by
sketching the cubic x3+ x2+ x 1 (by hand or by computer), you
should see thatthere is, in fact, exactly one zero in the
interval[0, 1]. Use Newtons method to nd this zeroaccurate to 3
decimal places. You should include a sketch of the cubic, Newtons
iterationformula, and the list of iterates. [Use a computer if
possible, e.g., a spreadsheet or MATLAB.]5. [5 marks] Using LH
opitals rule (or otherwise), determine the following limits.(a)
limx01 cos 3x22x2(b) limx1ln(4 3x)x 16. [23 marks] Evaluate the
following integrals, showing all your working.(a) x ln(1 + 2x2)
dx(b) 3164x2dx(c) 642s(s1)(s+2) ds(d) 21xx2+2x+5 dx(e)
12x2exdx[Hint: For the improper integral in part (e), you can use
the fact that exponentials grow faster thanpolynomials; in
particular, limx p(x)/ex= 0 where p(x) is a polynomial function of
x.]7. [4 marks] Compute the following integral, accurate to 3
decimal places, using any technique thatyou think is appropriate.
Show all your working. 120ex2dx
