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# Tutorial 8 - Suppose a particular species of insect grows...

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Tutorial 8 – Introduction to Calculus and Linear Algebra Page 1 of 1 Tutorial 8 Introduction to Calculus and Linear Algebra In Exercises 1 to 8, use the chain rule to compute the derivative and simplify your answer. 1. ; 1 2 + = u y 2 3 - = x u 2. ; u y = 3 2 2 - + = x x u 3. ; 1 2 u y = 1 2 + = x u 4. ; 1 u y = 9 2 - = x u 5. 4 ) 1 2 ( ) ( + = x x f 6. 5 3 ) 1 2 ( ) 2 ( ) ( - + = x x x f 7. 2 6 5 1 ) ( 2 + - = t t t f 8. 4 2 ) 1 ( 3 ) ( x x f - = In Exercises 9 to 11, find all values of x = c so that the tangent line to the graph of f ( x ) at (c, f ( c )) will be horizontal. 9. 2 2 ) ( ) ( x x x f + = 10. 2 ) 2 3 ( ) ( - = x x x f 11. 5 4 ) ( 2 + - = x x x f 12. In a research paper (Tucker and Schmidt-Koenig, 1971)*, it is demonstrated that a species of Australian parakeet expends energy according to the formula ] 22 ) 35 ( 074 . 0 [ 1 2 + - = v v E where v is the bird’s velocity (in km/h). Find a formula for the rate of change of E with respect to velocity v . 13. The growth of certain insects varies with temperature.
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Unformatted text preview: Suppose a particular species of insect grows in such a way that the volume of an individual is ) 52 . 3 4 . 01 . ( 41 . ) ( 2 + +-= T T T V cm 3 when the temperature is T o C, and that its mass is m grams, where V V V m 09 . 1 39 . ) ( + = (i) Find the rate of change of the insect’s volume with respect to temperature. (ii) Find the rate of change of the insect’s mass with respect to volume. (iii) When T = 10 o C, what is the insect’s volume? At what rate is the insect’s mass changing with respect to temperature when T = 10 o C? *V.A. Tucker and K. Schmidt-Koenig, “Flight Speeds of Birds in Relation to Energetics and Wind Directions, The Auk, Vol. 88, 1971, pp. 97-107....
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