Honors Calculus Homework 1, due 9/8/5

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1 Honors Calculus Homework 1, due 9/8/5 Question 1 Calculate the derivatives of the following functions: p(x) = x 4 3x x 4x q(x) = (1 + x)(1 + x 2 )(1 + x 3 )(1 + x 4 ). r(t) = (1 + t)(1 + t2 ) 1 + t 3. s(t) = t t 1 2. t t 1 4 Question 2 Calculate from first principles the derivatives of the following functions: f(t) = t 2 4t + 3. g(t) = 3 t + 1. h(t) = t 1 3. Question 3 Find the equation of the tangent and normal lines to the curve y = t t point of the curve with t = 1. Sketch the curve and the tangent and normal lines. Also find all points of the curve where the tangent line is horizontal. at the Question 4 Consider the limit: lim x = 2. x 4 Given ɛ > 0, find algebraically an open interval J ɛ containing 4, such that if x is in J ɛ, then x 2 < ɛ. Evaluate J ɛ in the cases that ɛ = 10 1, ɛ = 10 2 and ɛ = 10 3.

2 Question 5 Consider the limit: lim x 2 1 x 2 = 1 4. Given ɛ > 0, find algebraically an open interval J ɛ containing 2, such that if x is in J ɛ, then 1 x < ɛ. Evaluate J ɛ in the cases that ɛ = 10 1, ɛ = 10 2 and ɛ = Question 6 Construct a Maple routine (do loop) to produce the first 100 Fibonacci numbers. They are given by the recursion formula, valid for each integer n 2: f n = f n 1 + f n 2, with the starting values f 0 = 0 and f 1 = 1. For 2 n 100 compute the ratios r n = results. f n f n 1 and s n = f n 1 f n and discuss the 2

3 Honors Calculus Homework 2, due 9/15/5 Question 1 Calculate the derivatives of the following functions: x 3 x p(x) = x q(x) = x + x 1 x 3 + x 3. Question 2 A particle moving along a straight line has position x meters at time t > 0 secs given by the formula: Find its velocity and acceleration x = t 2 3t 1 t. Find all times at which the particle comes to rest. Find all times at which the acceleration of the particle is zero. Find the maximum speed of the particle on the time interval from t = 1 2 to t = 2 seconds. Question 3 Let f(x) = x. Find the equation of the tangent line to the graph y = f(x), at the point with x = 25. Find the linear approximation to the function f(x), based at the point x = 25. Plot the graph of the function and the linear approximation on the interval in x, [20, 30]. Use your linear approximation to estimate 27 and compare your estimate with the exact result. What is the percentage error in your estimate? 3

4 Question 4 For any given ɛ > 0, find an open interval J ɛ, containing x = 3, such that if x is in J ɛ, then x + 3 x 1 3 < ɛ. ( ) x + 3 Explain why this shows that lim = 3. x 3 x 1 Question 5 Let g(x) = x + 3 x 1. What is the domain of the function g(x)? By graphing the function g(x), or otherwise, determine its range. Show that the equation y = g(x) has a unique solution for x given y, unless y = 1. How many solutions are there to the equation y = g(x), when y = 1? Explain your answer. Question 6 A sequence x n is given by the recursion: x 1 = 1, and for any integer n 2 x n = 1 ( x n ). 2 x n 1 By implementing this recursion on Maple, or otherwise, discuss the behavior of this sequence for large n. If the initial value x 1 is changed to some other value than 1 does the large n- behavior of the sequence change? 4

5 Honors Calculus Homework 3, due 9/22/5 Question 1 Calculate the derivative of each of the following functions: a(t) = sin 2 (t 2 t + 1) b(x) = cos 3 (3x 3 ) c(t) = (1 + cos 2 (t)) 444 Question 2 A function f(x) has derivative 200x(1 x 2 ) 999. Given that f(0) = 0, what is f(2)? A function g(x) has derivative 900x 1001 (x + x 4 ) 999. Given that g(1) = 1, what is g(2)? Question 3 Functions f(t) and g(t) obey the following properties: f(2) = 2, f (2) = 3, g(4) = 2, g (4) = 3. Find the equation of the tangent line to the curve y = f(f(t)) at t = 2. Find the equation of the tangent line to the curve y = f(g(t)) at t = 4. Find the equation of the tangent line to the curve y = g(f 2 (t)) at t = 2. Question 4 Let h(x) = 2x 1 3x 2. What is the domain of the function h(x)? Find with proof the inversion function of the function h(x) and give the domains and ranges of h(x) and of its inverse. 5

6 Question 5 Write out the proof that lim x 9 x = 3. Question 6 A particle moving on the plane has position vector X given by the formula valid for 0 < t < 1: [ ] 1 X =, 2 ln(t) 2 ln(1 t). t(t 1) Find the velocity, acceleration and speed of the particle at time t. Find the minimum speed of the particle. 6

7 Honors Calculus Homework 4, due 9/29/5 Question 1 Calculate the derivative of each of the following functions: a(t) = ln(sin(t)) b(x) = arcsin(x 2 ) c(t) = arctan( 2t) p(x) = ln(x x 2 ) Question 2 Define a number c by the formula: 3 h 1 c = lim. h 0 h Show that c can be interpreted as the slope of the curve y = 3 x at x = 0. Show that: 9 h 1 2c = lim, h 0 h (27) h 1 3c = lim, h 0 h Show that the function f(x) = 3 x obeys the relation: f (x) = cf(x). By using appropriate numerical estimation, or by plotting the graph y = 3 x near x = 0, estimate the quantity c. Question 3 Write out the proof that lim x 5 x 2 = 25. 7

8 Question 4 Let f(x) = x3 x 2 x + 1. x + 2 Give the domain of f(x). Explain why the range of f(x) is all real numbers. Find the derivative of f and find all points where the graph y = f(x) has a horizontal tangent. Plot the graph of the function y = f(x). Find the largest domain for f of the form (a, ) for a suitable real number a, such that f has an inverse on that domain, explaining your answer. Question 5 Let g(t) = 1 8 (t3 12t). Plot the graph of the function g. Explain why the function g has an inverse if its domain is restricted to the interval J = ( 2, 2). What is the range K of g on the domain J? Explain. Let g 1 : K J be the inverse function. Find the equation of the tangent line to the function y = g(t) at the point with t = 1. Find the equation of the tangent line to the function y = g 1 (t) at the point with t = Sketch the functions y = g(t) and y = g 1 (t) and the two tangent lines on one graph, using the same scaling for each axis and discuss your results. 8

9 Question 6 Let A = [2, 5], B = [ 3, 7] and C = [14, 0]. Sketch the triangle ABC. Find the lengths of the sides of the triangle ABC. Find the angles at the vertices of the triangle ABC. Find the area of the triangle ABC. 9

10 Honors Calculus Homework 5, due 10/6/5 Prepare for the exam Friday: go through your notes and the material on our webpage, especially the quizzes and homeworks. Topics for the exam: derivatives from first principles, derivative formulas, tangent and normal lines, estimation, properties of functions, domain, range and inverse functions, velocity and acceleration. Question 1 Calculate the derivative of each of the following functions: a(t) = ln(ln(t 1010 )) b(x) = e ex c(t) = 2 t2 p(x) = ln r(x) = 2 34x Question 2 ( 3 x3) 3 ln(x3 ) Find the equation of the tangent and normal lines to the hyperbola x 2 4y 2 = 9 at the point (5, 2) of the hyperbola. Also sketch the hyperbola and your tangent and normal lines on one graph. Question 3 Let A = [3, 1], B = [4, 2] and C = [5, 2]. Sketch the triangle ABC. Find the lengths of the sides of the triangle ABC. Find the angles at the vertices of the triangle ABC. Find the area of the triangle ABC. Find the image of the triangle ABC under a counter-clockwise rotation through 45 degrees about the origin. 10

11 Question 4 Two vertical trees on a level plain are 50 meters apart and 30 meters high. A package of mass 40 kilos is suspended from the top of each tree by straight wires one of length 40 meters, the other of length 30 meters. Where is the package? Sketch the configuration. What are the tension forces in the wires? If the package is replaced by another of 100 kilos and the wires are required to have a 50 percent margin of safety before they break under the strain of supporting the package, what should be the least breaking strain of the wires? Question 5 Determine the point on the ellipse with equation 4x 2 + 9y 2 = 4 that is closest to the point (1, 4). Also determine the two tangent lines to the ellipse from the point (1, 4). Sketch the ellipse and the two tangent lines. Question 6 A particle moving in the plane has position vector X at time t given by X = [3 cos(2t), sin(3t)]. Find all points in the plane where its acceleration vector is perpendicular to its velocity vector. Also sketch the motion of the particle. What is the furthest distance of the particle from the origin, during its motion? Explain. What is the period of the motion? Explain. 11

12 Honors Calculus Homework 6, due 10/13/5 Question 1 Let A = [2, 3], B = [5, 4], C = [ 3, 1]. Find all points D such that the four points A, B, C and D form a parallelogram and sketch all the parallelograms on one graph. What is the area of each such parallelogram? Where is the centroid of each such parallelogram? Question 2 A circular disc of radius 2 meters rotates at 2 radians per second in a vertical plane, without slipping, with its lowest point in contact with a straight, horizontal line. A point P, initially in contact with the ground, is marked on the disc. Find formulas for the position vector X of the point P at time t seconds, its velocity vector V and its acceleration vector A. Find the maximum speed of the motion of P. Also plot the motion. Question 3 An ant is on the rim of a rotating disc of radius 10 centimeters. If the disc is rotating at 2 radians per second, find the velocity and acceleration of the ant. If now the ant starts walking radially towards the center of the disc, moving at 2 centimeters per second, relative to the disc, what now is its velocity and acceleration? Plot the motion of the ant. Question 4 Let A = [2, 3] and B = [8, 5]. Find the parametric equations of the line AB and of its perpendicular bisector. Also find the two unit vectors that make an angle of 45 degrees with the direction of the line AB. 12

13 Question 5 The period T of a pendulum of length L is given by the formula: L T = 2π g. Here g is the acceleration due to gravity. Use the linear approximation to relate the fractional change in the period to the fractional change in the length. In particular if the length is known an accuracy of within 0.1 percent, about how much possible error is there in the measurement of one day by the clock? 13

14 Question 6 A cylindrical can of volume 1.2 liters is made from material costing 40 cents per square meter for the sides and base of the can and 100 cents per square meter for the top. What are the dimensions of the can that minimize the total cost of materials for the can and what is that total cost? 14

15 Honors Calculus Homework 7, due 10/20/5 Question 1 Let A = [2, 3, 1], B = [3, 4, 2], C = [3, 1, 5] and D = [ 3, 3, 7] be points in space. Find the parametric equations of the line AB and the symmetric equations of the line CD. Show that the lines AB and CD meet and find their point of intersection. Find the angle between the lines AB and CD. Question 2 Let A = [4, 3], B = [2, 2], C = [3, 1], A = [0, 3], B = [5, 1] and C = [4, 2]. Sketch the triangles ABC and A B C. Show that the triangles ABC and A B C are congruent. Describe the unique Euclidean transformation R that maps the triangle ABC to the triangle A B C. Obtain the 3 3 matrix representation for R. Show that R is a rotation and find the center of the rotation. Question 3 A window frame is to be made with 20 feet of framing material. The shape of the window is to be a rectangle topped by a semi-circle, with no overlap. Find the dimensions of the window to maximize its area. 15

16 Question 4 Let f(x) = x 3 3x 2 24x Sketch the graph of f identifying its critical points, local maxima and minima and its inflection points. Prove that f has a root in the interval [0, 1] and locate the root to five decimal places, using Newton s method. Question 5 Find the maximum of the function x 2 + y 2, where (x, y) is a point on the ellipse x 2 + 3xy + 5y 2 10x 8y = 23. Interpret your result geometrically. Question 6 Boat Amelia starts from its harbor at noon and moves NNE at 20 knots. One hour later, boat Beatrice starts from a harbor 100 nautical miles due East of Amelia and sails NW at 30 knots. Where are the boats when they are at their closest and when how close do they get? 16