Math 171 Calculus I Spring, 2019 Practice Questions for Exam IV 1

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1 Math 171 Calculus I Spring, 019 Practice Questions for Eam IV 1 Problem 1. a. Find the linear approimation to f() = cos() at a = π. b. Use your approimation to estimate (π + 1/100) cos(π + 1/100) Problem. Multiple Choice. Peeling an orange changes its volume V, what does V represent? a. the volume of the rind b. the surface area of the orange c. the volume of the edible part Problem 3. Multiple choice: Suppose that f () < 0 for near a point a. Then the linearization of f at a is 1. an over approimation. an under approimation 3. unknown without more information. Problem 4. a. Find the linear approimation to a + for near zero. Here a is a positive constant. b. Use your approimation to estimate Problem 5. Suppose you are driving along a highway in a car with a broken speedometer at a nearly constant speed and you can record the number of seconds it takes to travel between two consecutive mile markers. If it takes 60 seconds, then your average speed is 1 mi/60 s or 60 mi/hr. The function s() = gives your average speed in mi/hr if you travel one mile in 60+ seconds. For eample, if you drive one mile in 6 seconds, = and your average speed is s() mi/hr. If you travel one mile in 57 seconds, then = 3 and your average speed is s( 3) mi/hr. Because you don t want to use your phone or calculator while driving, you need an easy approimation to this function. Use linear approimation to derive such a formula. Problem 6. Find the linear approimation for h 0 for w(h) = W (1 + h/r), which is the weight of a body at altitude h above the earth s surface, where W is the weight of the object of the body at the surface of the earth and R is the radius of the earth.

2 Problem 7. The quadratic approimation to a function f() at = a is given by Q() = f(a) + f (a)( a) + f (a) ( a). Find the quadratic approimation to f() = at = 4 and use it to approimate 3 8. Problem 8. Recall that Newton s method is used to find approimate solutions to f() = 0 by using the iteration: n+1 = n f(n) f ( n). Show how to setup Newton s method to find the cube root of b, i.e. show how would to setup Newton s method to compute a solution of 3 = b. Problem 9. Let (a, f(a)) be the point where the tangent line to the graph of f() = cos() is horizontal. We use Newton s method to find 1,, 3,..., the successive approimations to a. Give the formula that we use to compute n+1 from n. Problem 10. We will use each of the n below as the starting point for Newton s method. For which of them do you epect Newton s method to work and lead to the root of the function? 1. 1 and only.. only. 3. 1, and 3 only. 4. All four. Problem 11. Write the formula for Newton s method applied to f() = 1/5, and use the given initial approimation 0 = 1 to compute 1 and. What happens here? Does the method converge? Problem 1. Eplain why Newton s method eventually fails when finding zeros of f() = with a starting value 1 =. Problem 13. Find all critical points of the following functions (if they eist) a. f() = 3 3 b. f() = ( 4) 1 3 c. f() = e d. f() = 1 + e. f() = 1/3

3 Math 171 Calculus I Spring, 019 Practice Questions for Eam IV 3 Problem 14. Find the absolute maimum and absolute minimum values of the following functions on the given intervals. Specify the -values where these etrema occur. a. f() = on [, 3] b. h() = + 1 on [ 1, 1] c. f() = ln on [ e, e 4 ] d. f() = ln on (0, 1] (Hint: use L H rule to figure out what happens as 0 +.) Problem 15. The Reaction R() of a patient to a drug dose of size depends on the type of drug. For a certain drug, it was determined that a good description of the relationship is: R() = A (B ), 0 B where A and B are positive constants. The Sensitivity of the patients body to the drug is defined to be R (). a. For what value of is the reaction a maimum, and what is that maimum reaction value? b. For what value of is the sensitivity a maimum? What is that maimum sensitivity? Problem 16. Ecologists are often interested in the relationship between the area of a region (A) and the number of different species S that can inhabit that region. Hopkins (1955)suggested a relationship of the form [Hopkins, 1955] S = a ln(1 + ba) where a and b are positive constants. Does this function have a maimum? Eplain. Problem 17. Suppose that g() is differentiable for all and that 5 g () 3 for all. Assume also that g(0) = 4. Based on this information, use the Mean Value Theorem to determine the largest and smallest possible values for g(). Problem 18. The fastest drag racers can reach a speed of 300 mi/hr over a quartermile strip in 4.45 seconds (from a standing start), Complete the following sentence about such a drag racer: At some point during the race, the maimum acceleration of the drag racer is at least mi/hr/sec. Problem 19. Show that 1 + < for > 0. Problem 0. A trucker handed in a ticket at a toll booth showing that in hours she had covered 160 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why did she deserve the ticket?

4 4 Problem 1. The following graph is the graph of f (). Where do the points of inflection of f() occur and on which intervals is f concave up? Does f have a local ma or min at = 0? Problem. Sketch the graph of a function for which: f(0) = 0, f () > 0 for < 0, f () > 0 for > 0, lim f() = 1, f() = 0, lim f () > 0 for < 0, f () < 0 for > 0. Problem 3. Given that f() =, f () =, and f () = 6. Make +1 ( +1) ( +1) 3 a sketch of the graph the function f. Be sure to include local etrema and points of inflection. Problem 4. Find the interval(s) on which f is concave up or down, the points of inflection, the critical points, and local minima and maima for f() =. ln() Problem 5. Multiple Choice. The second derivative, f (), of a function f() is negative everywhere. We know that f(0) = 0 and f (0) = 0. What must be true about f(1)? a. f(1) is negative b. f(1) is positive c. f(1) is zero d. Not enough information to conclude anything about f(1). Problem 6. a. lim π π sin( π) b. lim sin c. lim 0 + ln() Evaluate the following limits. ( 1 ) d. + 1 lim + 1 e. 1 sin() lim π/ cos() f. lim ln() a ; a > 1. e h 1 g. lim h 0 h ln h. lim 1 i. lim 1 ( )

5 Math 171 Calculus I Spring, 019 Practice Questions for Eam IV 5 Problem 7. The bottom of the legs of a three-legged table are the vertices of an isosceles triangle with sides 5,5,6. The legs are to be braced at the bottom by three wires in the shape of a Y. What is the minimum length of wire needed? Show it is a minimum. B 3 A C Problem 8. In 1613, Kepler bought a barrel of wine for his wedding but questioned the method the wine merchant used to measure the volume of the barrel and thus determine the price. The price for a barrel of wine was determined solely by the length L of a dipstick that was inserted diagonally through a centered hole in the top of barrel to the edge of the base of barrel (see figure above). In consequence, afterwards Kepler set out to find the proportions that optimize the volume of such a barrel. Suppose that the wine barrel is a cylinder as shown. Determine the ratio of the radius r to the height h of the barrel that maimize the Volume of the barrel subject to the wetted length L of the dipstick is fied. Problem 9. An architect plans to build a rectangular window under a arch that is a semi-circle. Find the maimum area of a rectangle inscribed in semi-circle of radius r. y 0 r