Suppose f is a differentiable function that satisfies 2xf(x) + cos(f(x) - 1) =9 for all c. If f(4) = 1, what is f (4) ? Answer: f (4) = (You do not need to (nor can you!) find a formula for f(). Hint: differentiate both sides of the equation with respect to I.)The function y is given implicitly by the equation y" In(y) - 1 In(x) =6 Find the derivative of y as a function of r and y- Answer: y'A spherical snowball is melting in the sun. It is noted that its surface area decreases at a rate of 2 E1112 f 3 at the moment when its diameter is g cm. The goal here is to determine the late at which the diameter varies at that same moment. To solve this problemI let I be the diameter ofthe snowball in cm. A its surface area in m2 : and t the time in seconds (8]. (a) Express A as a function of :I: . [The surface area is a formula you can nd in yourtextbook.) A = m cm2 11A 5 (l1) What is the value of T when :I: = 'F' Give the exact value. 1: :rr IiA _ I E a \"m d: 5 (c) Using our previous results. give the (exact) value of E when :I: = cm. Beware of signs: remember that the surface area W of the snowball is decreasing with time! % = .acn'u's. A lcite glides horizontally at an attitude ot3m 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 40 m of string has been unsoooled. given that. at that instant. the kite's horizontal velocity is 1 ma's. To solve this problem, let 9 be the angle in radians made between the string and the horizontal. z the kite's horizontal position in meters since being attached to the ground. and f the time in seconds. We further suppose that the string is straight and taut. (a) Sketch a diagram ofthis question and use it to emress 3 as a function of I. H = Ia rao (b) What is the value of: at the moment in question? Give the exact value. I = a m d3 (c) What is the value of at the same moment? Give the exact value. paying attention to the sign. at}. E = a radfs 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 4 cm/s at the moment the triangle's area is 3v3 cm. 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 & the common length of the sides of the triangle in cm, A its area in cm" , and t the time in seconds (s). (a) Express A as a function of I . A = 40cm (b) What is r when A = 3\\/3 cm- ? Give the exact value. I = 60 cm. dA (c) What is when A = 313 cm- ? Give the exact value. dA Eicm (d) We know that dt =4 when A = 3V3. dA Using the chain rule, compute dt when A = 313 cm- . Give the exact value. dA dt cm/ sA projector, laying on the ground, illuminates the wall of a building standing 15 m away. A person 1.8 m tall walks from the projector to the wall. We would like to determine 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 1.1 m/s. To solve this problem, let z be the distance in meters between the walker and the wall, y the height of their shadow in meters, and t the time in seconds. (a) Sketch a diagram and use similar triangles to express y as a function of I . y (b) What is the value of dy dt at the moment in question? Give its value with two decimals precision, paying attention to the sign. dy Number m/s dtConsider the function g(I) = arcsin(e T) + 4. What is the derivative of g(x)? FORMATTING: Give an exact answer, not a decimal approximation. If needed: in Mobius, arcsin(x ) is written simply arcsin(x) and VI is written as sqrt(x). Answer: g' (x) =Consider the equation "+y' = 3ry +1. a) Use implicit differentiation to find the derivative of y with respect to c . Your answer will be a function of both I and y dy b) Now find the equation of the tangent line to the curve described by ro +y = 3ry + 1 at the point (0, 1) . Answer: FORMATTING: Your answer must be in the form of an equation for y in terms of ; e.g. y = ax + b