MAS111 Strand 1 Problems Spring Semester

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1 MAS111 Strand 1 Problems Spring Semester Week 1 (Review) 1. Recall that cos(a + B) = cos A cos B sin A sin B. educe an identity for cos(2), and use it to obtain epressions for cos 2 and sin 2. Evaluate the integral π/3 cos 2 2 d. 2. Let y = u/v, where u and v are (differentiable) functions of. Suppose that is changed by a small amount δ, with changes δu, δv, δy in response. Show that v δu u δv δy = v(v + δv). educe a formula for dy d. Assuming known formulas for the derivatives of sin and cos, deduce a formula for the derivative of tan. 3. Let y = tan 1 (inverse function). By differentiating both sides of = tan y, with respect to (so using the Chain Rule), obtain an epression for dy d. Use an appropriate trigonometric identity to write this purely in terms of. 4. ifferentiate e 2. Evaluate the integrals 1 e2 d and 1 3 e 2 d. 5. Evaluate the integrals 1 Week 2 d, d and d. 1. Solve the following inequalities and graph the solution sets: (a) 5 2 < 9 + 3; (b) t + 5 < 6; (c) 3y 9 < 4; (d) 2 6 < ; (e) ( + 3) 2 < 2; (f) 3 2/t < 1/2. 2. (a) By drawing the graphs, determine for which values of we have cos() > sin(), and for which values of we have cos() > sin(). 1

2 (b) Solve the inequality cos 2 2 sin cos sin 2 >. 3. Using the triangle inequality, prove that for any (real or comple) numbers a, b we have a b a + b a + b. [The left-hand inequality may be approached in the same spirit as 1(b).] 4. Graph the functions f() = /2 and g() = 1 + (4/) together to identify the values of for which 2 > Confirm your findings by solving the inequality algebraically. Week 3 1. Let f() = for 1. o a table of values of f() for some values of very close to 1. Prove algebraically that lim 1 f() is what it appears to be. 2. Let f() = o a table of values of f() for larger and larger values of. Prove algebraically that lim f() is what it appears to be. 3. o you think that lim 1/ eists? Plug in larger and larger values of to the formula 1/ and do a table. Now reconsider your opinion. 4. o you think that lim (1 + 1/) eists? Plug in larger and larger values of to the formula (1 + 1/) and do a table. Now reconsider your opinion. Can you prove anything? 5. Prove, from first principles, the formula for the derivative of cos(). You may, of course, assume any limit formulas proved in lectures. 6. Prove that if 1 then Can you improve the bound? 3+sin 7. raw a graph of the function f() = 2. Write down an epression for the slope of the chord joining the points (, 1) and (h, 2 h ) (h ) on this graph, using your picture for illustration. o a table of values of this slope for smaller and smaller values of h approaching zero. o 2

3 you recognise the apparent limit? Of which natural line on your picture is this the slope? Epress what you appear to have found as a statement about the value of a derivative. Answer the whole question! Week 4 1. Show that lim 2 sin 1 = by each of the following methods. (a) By using the Sandwich Theorem. (b) By using that lim sin 1 = and the rules of limits. 2. Find lim sin(7) by each of the following methods. (a) By using that lim sin() = 1. (b) By L Hopital s Rule. 3. (a) Show that 1 cos = sin 2 (1 + cos ) and deduce that lim 1 cos =. (b) Check this limit using L Hopital s Rule. 4. Calculate the following limits using l Hopital s Rule. (a) lim ; (b) lim 1 cos 2 ; (c) lim 1 e 2 ; (d) lim + ln. educe that lim + = What happens if you try to use l Hopital s Rule to find lim Find this limit another way. ( 2 + 1) 1/2? 6. Show that for any integer k, k e as. educe that this holds for any (not necessarily integer) k. 3

4 7. On the graph y = 1/t, draw a region whose area represents 1+1/ 1 1 t dt. By bounding this area between the areas of two rectangles on your ( picture, and evaluating the integral, obtain a new proof that lim ) = e. Week 5 1. Let f(, y) = 3 y 2 5e y. Calculate f and f y. 2. Sketch some level curves of each of the following functions. (a) f(, y) = y 2 ; (b) f(, y) = 2 + 2y 2 ; (c) f(, y) = y. 3. Sketch some level curves of the function f(, y) = (1/3)y 2 +. At the point (1, 1), according to your picture does f appear to be increasing more rapidly in the direction or in the y direction? Confirm your answer algebraically. 4. I walk along the surface 4z = 2 + (1/3)y 2 in such a way that increases and y stays constant. As I pass through the point (1, 1, 1/3), how steep is my path? 5. Show that f(, y) = e y cos is harmonic, i.e. 2 f f y 2 =. 6. raw some level curves for a function f(, y) in such a way that (a) f y (1, 4) > f (1, 4) > f f (1, 2) > y (1, 2) > ; (b) f f (3, 3) = y (3, 3) = ; (c) f(1, 2) = 2, f(1, 4) = 6, 1 < f(3, 3) < 11. There are many ways of doing this. You need to use your imagination. It is not necessary to write down a formula for f, only to draw some level curves consistent with, and illustrating, the above properties on an appropriate part of the -y plane. 4

5 Week 6 1. Let φ(, y) = 3 y + e y2. Find φ, φ y, 2 φ 2, 2 φ y, 2 φ y Let u = 2 + y and v = 2y. Evaluate the Jacobian (u, v)/ (, y). Then epress and y in terms of u and v. Hence evaluate the Jacobian (, y)/ (u, v) and confirm that it is the reciprocal of the previous answer. 3. For > y > let u = + y + y and v = + y y. Evaluate the Jacobian (u, v)/ (, y) (which in this case is always positive). Then epress and y in terms of u and v. Hence evaluate the Jacobian (, y)/ (u, v) and confirm that it is the reciprocal of the previous answer. 4. The radius of a circular cylinder goes up by.3% while its height goes down by.2%. Estimate the percentage change in volume. 5. Find an equation of the tangent plane to the surface z = 2 + y 2 at the point where = 1 and y = Let z = y 2 + y 4, where = r cos θ and y = r sin θ. Use the Chain Rule to show that z/ r = 4r 3 and z/ θ =. Then, as an alternative method, epress z in terms of r and θ and obtain these answers more directly. 7. (a) Show that if f has equal mied second order partial derivatives and g(, y) satisfies g = f y and g y = f then g is harmonic. (Recall this means that g satisfies 2 g g y 2 =.) 5

6 (b) Find a function g that is related in this way to the function f given by f(, y) = e y cos. Week 7 1. Let z = f(, y), where = u + v and y = u v. Show that 2. If y is defined implicitly by z z ( z ) 2 ( z ) 2. u v = y 2 y + sin( + y) =, epress dy/d in terms of and y. 3. Let z be a function of u and v, where u = 2 + y 2 and v = y. Show that and that 2 z 2 z = 2 z u + y z v, = 2 z u z u 2 + 4y 2 z u v + y2 2 z v 2. Hence write down a similar epression for 2 z. Use these to obtain an y 2 epression for 2 z + 2 z which involves only u, v and various partial 2 y 2 derivatives of z with respect to u and v. (You may assume equality of mied second-order partial derivatives.) 4. Aim: to find the general solution to the Wave Equation: 2 y t 2 = c2 2 y 2. [This is a partial differential equation governing small displacements y of a taut string, where is position along the string, t is time and c 2 = tension/mass per unit length.] Idea: change variables to u = ct, v = + ct. Show that y = y u + y y v, and similarly obtain an epression for t. Using [e.g. = 2 y 2 u + v, etc., epress 2 y and 2 y 2 t 2 = 2 y y u 2 u v + 2 y ]. v 2 6 in terms of 2 y u 2, 2 y v 2, 2 y u v,

7 Hence show that the wave equation boils down to 2 ( ) y u v =, i.e. y =. u v Integrating twice, deduce that the general solution is y(, t) = f( ct) + g( + ct) for any f and g. This is the superposition of a wave (shaped like the graph of f) travelling in the positive direction with speed c and a wave (shaped like the graph of g) travelling in the negative direction with speed c. Week 8 1. Let S be the square {(, y) : 1 2, y 1}. (i) Evaluate ( 2 3y 2 ) d dy. (ii) Evaluate S S 1 ( + y 2 d dy. ) 2 2. Let be the region of the plane {(, y) :, y, + y 1}. Sketch the region in the plane. Evaluate the following integral either first with respect to and then with respect to y, or else the other way round: ( 2 3y 2 ) d dy. (Check with someone who did the integration the other way that the answers are the same.) 3. Let R be the triangular region of the (, y)-plane with the vertices (, 1), (1, 2) and (2, 1). Evaluate 2 + y + 2 d dy. + y The integral R 1 y e (3 3) d dy at first sight is impossible. Epress it as a double integral over a region. Sketch. Evaluate the double integral by calculating the iterated integral in which the y-integral is done first. 7

8 Week 9 1. Let a >. By changing the order of integration in the repeated integral prove that a ( a ( 2 + y 2 ) dy ) d = 1 2 ( 2 1)a By changing the order of integration in the repeated integral, evaluate 1 1 y 2 y 9 (1 6 ) d dy. 3. Let be the region in the first quadrant bounded by the curve 2 y 2 = 1/4 and the lines = 1, y =. Evaluate y d dy. ( ) 4. Let be the region in the (, y)-plane for which 2 + y 2 < 1, y. By changing to polar coordinates show that ( + y) ( 2 + y 2 ) 2 d dy = π ln The formula for the area of a surface z = f(, y) over a region is ( z ) 2 ( z ) d dy. y By considering that part of the sphere 2 + y 2 + z 2 = R 2 which lies above the set = {(, y) :, y, 2 + y 2 R 2 } use the above formula to show that the surface area of a sphere of radius R is 8 R d dy. R 2 2 y2 By changing to polar coordinates confirm that this surface area is 4πR 2. 8

9 6. Let be the region {(, y) :, y e and y 2 2}. Let u = 2 + y 2, v = ye. Evaluate the Jacobian (u, v)/ (, y) and use it to find the double integral ( + y 2 )e 2 + y 2 d dy Let be the region in the first quadrant of the (, y)-plane bounded by the curves y = 2, 2y = 2, = y 2, 2 = y 2. Sketch. By using the substitution u = 2 /y, v = y 2 / prove that ln(y) d dy = 2 (2 ln 2 1) The triangle is bounded by the lines y = 1/2, = y, 3y = + 2. Sketch. By using the substitution u = y, v = + y, evaluate ( y)(2 2y + 1)e 2 y 2 d dy. 9. Let be the region in the first quadrant bounded by the curves y = 1, y = 2, 2 + y 2 = 6, 2 + y 2 = 8 for which > y. Evaluate 2 y 2 ( 2 + y + y 2 d dy ) 2 by making the substitution u = y, v = 2 + y 2. 9

10 Week Consider the infinite series n=1 1. We know that this converges to n 2 something. Calculate the first seven partial sums (use a calculator). How big is the net term to be added on to get the eighth partial sum? oes this mean that we know n=1 1 correct to 1 decimal n 2 place? Work out a few more partial sums and reconsider your answer. 2. Consider the infinite series oes this series converge absolutely? Let s n be the sum of the first n terms of this series. Calculate the first ten of these partial sums, to 4 decimal places. What do you notice about the sequence s 1, s 3, s 5,..., and what do you notice about the sequence s 2, s 4, s 6,...? Can you eplain this? What do you notice if you compare the odd partial sums with the even partial sums? o you think they converge to a limit? If so, how many terms of the infinite series would you have to add up to know that you had the sum correct to 4 decimal places? Are you sure? 3. Prove carefully that the infinite series n=1 cos n converges. (You may n 2 assume anything proved in lectures if you state it clearly.) 4. Calculate the Maclaurin series for ln(1 + ) and find its radius of convergence. oes it converge at the endpoints of the interval of convergence? 5. Calculate the radius of convergence of the series (n!) 2 n=1 (2n)! n. Is it easy to tell whether or not it converges at the endpoints of the interval of convergence? 6. Can you calculate 1 sin 1 d? Approimating sin by the sum of the first three non-zero terms of its Maclaurin series, obtain an approimate value. Try again using the first four. Can you be sure how accurate your approimation is? 1