Question 1 Let y=272 3x. 323 5x2. 72r+3 a) Given that In (y) can be written in the form In (y) = In(A)z + In (B) + In (C) z+ In (D). What are the coefficients A, B, C, and D ? A = B = C= D - b) Using the correct answer to (a), and implicit differentiation, find the value of ay when I = 0. Answer - Question 2 Find the equation of the tangent line to the curve at the point (1, 5) . (Hint: simplify before differentiating ) Your answer must be an equation of the form y = me + b This question accepts couations. E.g. y-2 - 5(x-4)+1. - Question 3 f(z) = In(2x +1) , then its fourth derivative is f (4) (z) =Question 4 Evaluate In (2 + h) - In (2) lim h-+0 h - Question 5 Consider the function f(z) = 3arcsin(x) arccos(x) - 6arctan(x). What is the derivative of f(x) at z = 07 Answer: f (0) - FORMATTING: Your answer should be exact. Remember to write symbols like w and vI as pi and sqri(x) , respectively- Question 6 Match the following numbered functions with their derivatives: sec( I) desc(z) d tan(I) cot(I) arcsin(I) d -arcoos(I) darctan(I) 1. sec (I) 2. 1+2 3. V1-12 4. - cot(z) csc(I) 5. tan(x) sec(I) 6. 7. - csc (I)Question 7 Consider the function f(z) = (arctan(z))* + 3arctan(z) + 3. What is the derivative of f(z)? FORMATTING: In Mobius, arctan(z) is written arctan(x) Answer: f' (x) - Question 8 Consider the function o(z) - arcsin(e ) + 4. What is the derivative of g(x)? FORMATTING: Give an exact answer, not a decimal approximation. If needed: in Mobius, arcsin(2) is written simply arcain(x) and ,/2 is written as aqrt(x) Answer. g' (x) = - Question 9 Consider the equation 13+3 =3ry + 1. a) Use implicit differentiation to find the derivative of y with respect to c . Your answer will be a function of both r and y dy b) Now find the equation of the tangent line to the curve described by 23 ty - 3xy + 1 at the point (0, 1) Answer: FORMATTING: Your answer must be in the form of an equation for y in terms of z; e.g. y = ax + b.- Question 10 Suppose f is a differentiable function that satisfies 2 f(z) + cos(f(z) - 2) - 13 for all r. If f(3) = 2. what is f' (3) ? Answer: f (3) - (You do not need to (nor can you!) find a formula for f(). Hint: differentiate both sides of the equation with respect to r.)- Question 11 The function y is given implicitly by the equation yo In(y) - 25 In(z) = 5 Find the derivative of y as a function of r and y- Answer: y'= Question 12 A spherical snowball is melting in the sun. It is noted that its surface area decreases at a rate of 4 cm- / s at the moment when its diameter is ~ cm. The goal here is to determine the rate at which the diameter varies at that same moment. To solve this problem, let x be the diameter of the snowball in cm, A its surface area in cin" , and f the time in seconds (s). (a) Express A as a function of I. (The surface area is a formula you can find in your textbook.) (b) What is the value of -- when I = dA 7 Give the exact value. dA cm (c) Using our previous results, give the (exact) value of - dt when = = - em. Beware of signs, remember that the surface area of the snowball is decreasing with time! dr DE cmis.Question 14 An animated short film shows an equilateral triangle whose dimensions vary with time. Assume the triangle's sides have an instantaneous rate of growth of 7 cmis at the moment the triangle's area is 313 cme . The goal is to determine at what rate the area of the triangle is growing at that same moment To solve this problem, let's denote by a the common length of the sides of the triangle in cm, A its area in cine , and & the time in seconds (s). (a) Express A as a function of z . A- (b) What is I when A - 3:3 cm- 7 Give the exact value I = 2 cm. dA (c) What is when A - 3v3 cm- 7 Give the exact value. d.A (d) We know that = =7 when A = 3,/3 dA Using the chain rule, compute - when A - 31/3 cm . Give the exact value dA Question 15 A projector, laying on the ground, illuminates the wall of a building standing 15 m away. A person 2 m tall walks from the projector to the wall. We would like to delermine the rate at which the person's shadow decreases the moment the person is I'm from the wall, given that, at that moment, the person is walking toward the wall at 0.9 m/s. To solve this problem, let a be the distance in meters between the walker and the wall, y the height of their shadow in meters, and & the time in seconds. (a) Sketch a diagram and use similar triangles to express y as a function of I . (b) What is the value of dy at the moment in question? Give its value with two decimals precision, paying attention to the sign. dy Number m/s- Question 13 A kite glides horizontally at an altitude of 30 m while we unspool the string. Consequently, the angle made between the string and the horizon diminishes. We would like to determine the rate at which this angle decreases once 60 m of string has been unspooled, given that, at that instant, the kite's horizontal velocity is 3 m/'s. To solve this problem, let / be the angle in radians made between the string and the horizontal, z the kite's horizontal position in meters since being allached to the ground, and & the time in seconds We further suppose that the string is straight and taut. (a) Sketch a diagram of this question and use it to express & as a function of I . rad (b) What is the value of r at the moment in question? Give the exact value. de (c) What is the value of de at the same moment? Give the exact value, paying attention to the sign. de rad's