Tutorial 7 - Tutorial 7 – Introduction to Calculus and...

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Unformatted text preview: Tutorial 7 – Introduction to Calculus and Linear Algebra Tutorial 7 Introduction to Calculus and Linear Algebra In Exercises 1 to 8, differentiate the given function. 1. 3. 5. 7. 1 1⎞ ⎛ f ( x) = ( x 5 − 2 x 3 + 1)⎜ x − ⎟ 3 x⎠ ⎝ t f (t ) = 2 t −2 x 2 − 3x + 2 f ( x) = 2 2 x + 5x − 1 f ( x) = (2 + 5 x) 2 2. 4. 6. 8. x +1 x−2 f ( x) = 3 x+5 (2 x − 1)( x + 3) f ( x) = x +1 2 t + t g (t ) = 2t + 5 y= In Exercises 9 to 11, find all points on the graph of the given function where the tangent line is horizontal. f ( x) = ( x + 1)( x 2 − x − 2) x +1 11. f ( x) = 2 x + x +1 9. 10. f ( x) = x 3 ( x − 5) 2 In Exercises 12 and 13, find the rate of change dy/dx for the prescribed value of x. 12. y = ( x 2 + 3)(5 − 2 x 3 ) at x = 1 13. y = x+ 3 2 − 4x at x = 0 In Exercises 14 and 15, find the second derivative of the given function. 14. y = 2 5 x − 4x3 + 9x 2 − 6x − 2 5 15. y = ( x 3 + 2 x − 1)(3x + 5) 16. A large city commissions a study that indicates that spending money on pollution control is effective up to a point but eventually becomes wasteful. Suppose it is known that when x million dollars is spent on controlling pollution, the percentage of pollution removed is given by 100 x P ( x) = 0.03x 2 + 9 (i) At what rate is the percentage of pollution removal P(x) changing when 1 million dollars are spent? Is the percentage increasing or decreasing at this level of expenditure? (ii) For what values of x is P(x) decreasing? Page 1 of 2 Tutorial 7 – Introduction to Calculus and Linear Algebra 17. After t hours of an 8-hour trip, a car has gone D(t ) = 64t + 10 2 2 3 t − t kilometers. 3 9 (i) Derive a formula expressing the acceleration of the car as a function of time. (ii) At what rate is the velocity of the car changing with respect to time at the end of 6 hours? Is the velocity increasing or decreasing at this time? (iii) By how much does the velocity of the car actually change during the seventh hour? Page 2 of 2 ...
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