A can f 3 ever be negative give a reason for your

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Applied Calculus
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Chapter 5 / Exercise 12
Applied Calculus
Berresford/Rockett
Expert Verified
(a) Can f (3) ever be negative? Give a reason for your answer. (b) Can f ( - 3) ever be negative? Give a reason for your answer. (c) How large can f (3) be? How large can f ( - 3) be? (d) Must f have a critical point between - 3 and 3? Either show your reasoning or give a counterexample. 86
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Applied Calculus
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Chapter 5 / Exercise 12
Applied Calculus
Berresford/Rockett
Expert Verified
Math 180 Review Worksheets R3 8. Di erentiate the following functions, you do not need to simplify your answers. (a) f ( x ) = 5 p x 8 (b) f ( x ) = ln( x 2 - 4) (c) f ( t ) = arctan 2 t 2 (d) f ( x ) = x 2 tan(2 x 3 + 1) (e) f ( x ) = x 3 x 87
Math 180 Review Worksheets R3 9. Use the Intermediate Value Theorem to show that the polynomial f ( x ) = x 2 + x - 1 has a zero in the interval (0 , 1). 10. Find the shortest distance from the point (4 , 0) to a point on the parabola y 2 = 2 x . Hint: Minimize the distance squared. 88
Math 180 Review Worksheets R3 11. Consider the function f ( x ) = x 1 / 3 . (a) Determine the absolute extrema of f on [ - 1 , 8]. (b) Find the intervals where f is increasing and where it is decreasing. (c) Find the intervals where f is concave upward and where it is concave downward. (d) Sketch the graph of f on the grid below. (e) Why does the Mean Value Theorem fail for f on [ - 8 , 8]? 89
Math 180 Review Worksheets R3 12. Let f ( x ) = x 2 - x - 2 x 2 - 2 x + 1 = x 2 - x - 2 ( x - 1) 2 , f 0 ( x ) = 5 - x ( x - 1) 3 and f 00 ( x ) = 2( x - 7) ( x - 1) 4 . (a) Find the following limits. i. lim x !1 f ( x ) ii. lim x !-1 f ( x ) iii. lim x ! 1 - f ( x ) iv. lim x ! 1 + f ( x ) (b) Does f have any asymptotes? (c) On what intervals is f increasing? decreasing? At what values of x does f have a local maximum? local minimum? 90
Math 180 Review Worksheets R3 (d) On what intervals is f concave upward? concave downward? Does f have any points of inflection? (e) Sketch a graph of f using information from the previous parts. (f) Does f have an absolute maximum value? absolute minimum value? 91
Math 180 Review Worksheets R3 13. Find the linear approximation L ( x ) of the function f ( x ) = ln( x ) at a = 1 and use it to approximate ln(9 / 10). 14. Find the equation for the tangent line at the point (1 , - 1) of the curve given implicitly by: x 2 y - 3 y 3 = x 2 + 1 . 92
Math 180 Review Worksheets R3 15. Suppose f is a function that is continuous and di erentiable everywhere with the following properties: f has a horizontal asymptote at y = 1 f 0 ( x ) > 0 on the interval ( -1 , - 2) and (0 , 1 ), and f 0 ( x ) < 0 on the interval ( - 2 , 0) f 00 ( x ) > 0 on the interval ( -1 , - 3) and ( - 1 , 1 ), and f 00 ( x ) < 0 on the interval ( - 3 , - 1) f ( - 3) = 2, f ( - 2) = 3, and f ( - 1) = 2 Sketch the graph of f on the grid below. 16. A paper cup shaped like a cone with height 20 cm and top radius 4 cm springs a leak and water begins to pour out through the bottom. If the water is flowing out at a rate of 2 cm 3 /second when the water level is at the halfway point, how fast is the water level decreasing at this time? 93
Math 180 Review Worksheets R3 17. Consider the integral Z 4 - 4 1 - x 2 dx. Use a right Riemann sum approximation with 4 subintervals to estimate the value of this integral.

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