1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2

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1 Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form, i.e, no decimal approximations. 3. You may use calculators, but no graphing calculators are allowed. Good luck! 1. Compute the derivatives of the following functions, by any means necessary. x 1 a) fx) = 1 x 3 We use the Quotient Rule: f x) = 1 x3 )1/)x 1) 1/ x) x 1 3x ) 1 x 3 ) b) rt) = sin e 4t) We use the Chain Rule, as the function is a composition: r t) = cos e 4t) e 4t 8t) c) gx) = arctan 4 + x) We use the Chain Rule, keeping in mind the derivative of arctanx): g x) = x) 1/)4 + x) 1/ 1) d) hv) = Kv sec1 + v 3 ) We use the Product Rule, treating K as a constant, and remembering the derivative of secx): h v) = Kv sec1 + v 3 ) tan1 + v 3 ) + v sec1 + v 3 )) e) wθ) = tanθ) θ Since we have a variable in the base and a variable in the numerator, we need to use logarithmic differentiation: 1

2 Let y = tanθ) θ. Then, after taking a natural logarithm of both sides, we have lny) = ln tanθ) θ) lny) = θ ln tanθ)) y y = θ sec θ + 1) lntan θ) tan θ ) y = θ sec θ + 1) lntan θ) tan θ) θ tan θ 1)

3 . Determine the x values of all points on the ellipse 9x 1) + 4y + 3) = 36 that have horizontal tangent lines. We need to find the x values at which the derivative dy/dx is 0. Finding dy/dx requires implicit differentiation: 18x 1) + 8y + 3) dy dx = 0 dy 18x 1) = dx 8y + 3) Now, this derivative is 0 only if the numerator is 0. So, we solve implying that x = 1 3. Using implicit differentiation, find dy dx 18x 1) = 0, if x and y satisfy the equation e x/y = x y. ) ) y xy e x/y y = 1 y e x/y y xy ) = y y y e x/y y e x/y xy = y y y y y e x/y xy = y e x/y y ) y y e x/y x = y e x/y y y = y e x/y y y e x/y x 3) 3

4 4. The number if yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n = ft) = a 1 + be 0.7t. a) According to this model, what value does the population approach as t? If t, then the quantity 0.7t, forcing the expression be 0.7t to approach 0. Hence the population ft) approaches a. b) What is the rate of growth of the population as a function of t? We compute the derivative dn/. We could use the Quotient Rule, but it s a bit easier to re-write the function as ft) = a 1 + be 0.7t) 1. Differentiating using the Chain Rule, we have f t) = a 1 + be 0.7t) be 0.7t ) 0.7) 5. Find the equation of the tangent line to the curve gw) = log 5 w + 1) when w = 0. We first need the derivative: g w w) = w + 1)ln 5). The slope of the tangent line at w = 0 is m = g 0) = 0, and also g0) = 0. Thus x 1, y 1 ) = 0, 0). Putting this into point-slope form y y 1 = mx x 1 ), we have 6. Find t=1 if rt) = t + t + t. First re-write the function as rt) = Using the Chain Rule several times, we have y = 0 t + t + t 1/) ) 1/ 1/. = 1 t + t + t 1/) ) 1/ 1/ t + t 1/) 1/ )) t 1/ But, we re after t=1, so we need to substitute 1 in for t above: = /) ) 1/ 1/ /) 1/ )) 1 1/ 4

5 7. Newton s Law of Gravitation says that the magnitude F of the force exerted by an object of mass m on an object of mass M is F = GmM r, where G is the universal gravitational constant and r is the distance between the objects. a) Find df/ and explain its meaning. First let s re-write the function as F = GmMr. Then, df = GmMr 3 The represents the rate of change of the magnitude of the force exerted as the distance between the objects increases. b) What does the minus sign indicate? The minus sign indicates that the force exerted decreases as the distance between the objects increases. 8. A stone is opped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after 3 seconds. We use the formula for the area of a circle: A = πr. Note that we re looking for da/ not da/) when t = 3. So we differentiate implicitly with respect to t: da = πr. Now, the ripple is traveling outward at a speed of 60 cm/s. This means the rate of change of the radius is 60. So = 60. Also, what is the radius when t = 3? Since the radius is increasing by 60 cm every second, that means the radius is 180 cm after 3 seconds. So, r = 180 when t = 3. Plugging this into our equation for da, we have da = π180)60) = 1600π, where the units are in square centimeters per second. 5