#$&* course MTH 271 04/077pm 018. `query 18
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Given Solution: `a This function can be expressed as f(g(x)) for g(x) = x^3-4 and f(z) = 3 / z^2. The 'inner' function is x^3 - 4, the 'outer' function is 3 / z^2 = 3 z^(-2). So f ' (z) = -6 / z^3 and g'(x) = 3x^2. Thus f ' (g(x)) = -6/(x^3-4)^3 so the derivative of the whole function is [3 / (x^3 - 4) ] ' = g ' (x) * f ' (g(x)) = 3x^2 * (-6/(x^3-4)^3) = -18 x^2 / (x^3 - 4)^3. DER** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q **** Query problem 2.5.62 tan line to 1/`sqrt(x^2-3x+4) at (3,1/2) **** What is the equation of the tangent line? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This equation can be written as two functions and their derivatives: 1) = 1/sqrt(z) OR z^-0.5 1’) = -0.5(z^-1.5) 2) = x^2 - 3x + 4 2’) = 2x - 3 the complete derivative is found using the chain rule: (1’)*(2’) → (-0.5)*(z^-1.5)*(2x - 3) → (-0.5)*(2x - 3)/ sqrt(z)^3 → (-x + 3/2) / (sqrt( x^2 - 3x + 4)^3) When x = 3, y = -3/16 So this makes the equation: (y - 0.5) = -3/16 *(x -3) → y = -3/16 x + 17/16 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The derivative is (2x - 3) * -1/2 * (x^2 - 3x + 4) ^(-3/2) . At (3, 1/2) we get -1/2 (2*3-3)(3^2- 3*3 + 4)^(-3/2) = -1/2 * 3 (4)^-(3/2) = -3/16. The equation is thus ( y - 1/2) = -3/16 * (x - 3), or y = -3/16 x + 9/16 + 1/2, or y = -3/16 x + 17/16. DER** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q **** Query problem 2.5.68 rate of change of pollution P = .25 `sqrt(.5n^2+5n+25) when pop n in thousands is 12 **** At what rate is the pollution changing at the given population level? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This can be written as two functions and their derivatives: 1) = (0.5n^2 + 5n + 25) and 1’ = (n + 5) 2) = (z^0.5) and 2’ = 0.5*(z^ -0.5) Using the chain rule, the complete derivative is: (0.25) [(n+ 5) * (0.5) *(z^ -0.5)] → (0.25) [(n+ 5) * (0.5) *(0.5n^2 + 5n + 25)^ -0.5] → (0.25) *[ (n + 5)* ( 0.5) / ( sqrt(0.5n^2 + 5n + 25))] → (0.125)(n + 5) / (sqrt(0.5n^2 + 5n + 25)) or also: (n + 5) / (8*sqrt(0.5n^2 + 5n + 25)) When n = 12, P = 0.17 confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The derivative is .25 [ (n + 5) * 1/2 * (.5 n^2 + 5 n + 25) ^(-1/2) ) = (n+5) / [ 8 `sqrt(.5n^2 + 5n + 25) ] When n = 12 we get (12+5) / ( 8 `sqrt(.5*12^2 + 5 * 12 + 25) ) = 17 / 100 = .17, approx. DER** ADDITIONAL COMMENT Details of calculating P ': P is of the form f(g(x)) with g(x) = .5 n^2 + 5 n + 25 and f(z) = .25 z^(1/2). g ' (x) = n + 5 and f ' (z) = .25 ( 1/2 z^(-1/2) ) = 1 / (8 z^(1/2)), or -1 / (8 sqrt(z) ). Thus P ' = g ' (x) * f ' (g(x)) = (n + 5) * (1/ (8 sqrt(n^2 + 5 n + 25) ) ) = (n+5) / (8 sqrt(n^2 + 5 n + 25). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: 3 " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!