แคลคูลัสสำหรับการเงินระดับมหาวิทยาลัยcalculus_finance
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The document comprises a series of calculus exercises, focusing on functions, limits, derivatives, integrals, and Taylor series expansions. It includes specific problems such as finding inverses, calculating limits, and differentiating various functions. Additionally, it involves evaluating expressions and transformations related to multivariable functions.
แคลคูลัสสำหรับการเงินระดับมหาวิทยาลัยcalculus_finance
- 1. Mathematics Primer Exercises 1 Calculus Problem Sheet 1. Consider two functions f (x) = 9x + 2 and g (x) = x 9 2 9 : Show that they are functions of one another. 2. Obtain the inverse of the function f (x) = x1=3 + 2: 3. Calculate the following limits: lim x!2 x2 4 x 2 lim x!1 x2 x 2x2 + 5x 7 lim x! 25 p x + 5 x 25 lim h!0 (x + h) 3 x3 h lim h! 2 h3 + 8 h + 2 lim t!1 (1=t) 1 t 1 lim x! p 2 x2 + 3 (x 4) 4. Using the de…nition of the derivative, show that for y = 2x + 1; y0 = 2 f (x) = 1 x 2 ; f0 (x) = 1 (x 2) 2 g (x) = jx 5j ; no derivative exists at x = 5 5. Di¤erentiate the following functions y; to obtain dy dx where : y = x2 4x + 2 5 y = 1 (4x2 + 6x 7) 3 y4 + 3y 4x3 = 5x + 1 y = ln 3 q (2x + 5) 2 y = cos (4 3x) y = x2 exp(x) y = 3x2 x + 2 4x2 + 5 6. Calculate the following Z p x x2 4x + 2 dx Z 1 4 (3 p x + 1) ( p x 2) dx Z 2 1 2s 7 s3 ds Z 2 3 x2 1 x 1 dx Z 5 1 j2x 3j dx Z 5x 12 x (x 4) dx 7. By using suitable substitutions (change of variable), evaluate the following Z 3 x4 3 x3 dx Z x2 + x (4 3x2 2x3) 4 dx Z ( p u + 3) 4 p u du Z 1 + 1 u 3 1 u2 du Z x exp x2 dx Z sin x exp (cos x) dx 1
- 2. 8. If f (x; y) = (x y) sin (3x + 2y) ; determine fx; f y; fxx; fyy ; fxy; fyx: Now evaluate these expressions at (0; =3) : 9. Show that z = ln (x a) 2 + (y b) 2 satis…es @2 z @x2 + @2 z @y2 = 0 except at (a; b) : 10. Obtain Taylor series expansions for the following functions about the given point x0: If no point is given, then expand about the point 0 (in which case you can use standard Taylor series expansions) f (x) = x2 sin x f (x) = cos x; x0 = =3 f (x) = exp x; x0 = 3 f (x) = 1 1 4x f (x) = 3 2x + 5 f (x) = x2 + 1 x 1 11. If U (x; y; z) = 2x2 yz + xz2 ; where x = 2 sin t; y = t2 t + 1; z = 3 exp ( t) ; …nd dU dt at t = 0: 12. Given w = f (x; y) ; x = r cos ; y = r sin ; show that @w @r 2 + 1 r2 @w @ 2 = @w @x 2 + @w @y 2 2