PRACTICE FINAL - MATH 1210, Spring 2012 CHAPTER 1

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1 PRACTICE FINAL - MATH 2, Spring 22 The Final will have more material from Chapter 4 than other chapters. To study for chapters -3 you should review the old practice eams IN ADDITION TO what appears here. Any problem from either source is fair game. Also there may be some true/false problems that don t appear eactly as they do here. Just make sure you know the main theorems and when you are/aren t allowed to use them. The point is, I will try to put a lot of problems on the eam, but they will all be short ones. CHAPTER. True/False. Eplain your answer. (a) (b) a (2 + 3) = a a 2 = ( 2 ) ( ) (c) Limits can pass inside and outside of cube root symbols. (d) It is possible for a function to be continuous at = and the it at = DNE. (e) It is possible for a function s it to eist at = but not be continuous at =. (f) If f() and g() are both continuous everywhere then f(g()) is continuous everywhere. (g) If a function s one sided its eist at = then the it at = eists. (h) sin = (i) All polynomials are continuous everywhere. (j) f() = is continuous everywhere. (k) 2 =

2 CHAPTER 2. True/False. Eplain your answer. (a) If f (c) eists then f() is continuous at = c. (b) If f() is continuous at = c then f (c) eists. (c) If f (c) eists then f (c) also eists. d (d) (f()g()) = f ()g () for differentiable functions f, g. d (e) If f is differentiable everywhere then f (c ) = c f () for any constant c. (f) If an object s acceleration a(t) >, then velocity v(t) is increasing. (g) If an object s acceleration a(t) >, then velocity v(t) >. (h) If an object s position s(t) is increasing then its acceleration a(t) >. (i) If dv/dr = 5 and dr/dt = 2 then dv/dt = (j) d d (3 + 2)5 = 5(3 + 2) 4 2. Implicitly differentiate 3 + y 3 = 3y. You do not need to solve for y afterwards. 3. Assuming a function y satisfies y = 3, find the equation of the tangent line to the 2y curve through the point (3, 2). 4. If a balloon s volume is V = 4 3 πr3, find an epression for how fast the volume is increasing with respect to time, in terms of r and dr/dt. 5. Find dy if y = 3 2 +, = 2, and d =.. CHAPTER 3. True/False. Eplain your answer. (a) Every critical point is a place where f () = or f () DNE. (b) Every critical point is a local minimum or maimum. (c) If a function is continuous on a closed interval it achieves its global maimum. (d) If f () changes from negative to positive at = 3 and f (3) =, then there is a local minimum at = 3. (e) All points of inflection occur at a critical point. (f) For f() = 2 6 f(5) f(2), there is a point = c in the interval [2, 5] at which = 5 2 f (c). (g) The local etreme for f() = 3 3 at = 3 is a local minimum. 2

3 (h) If F () and G() are both antiderivatives of f() then F () = G() + C for some constant C. (i) If f() and g() both have antiderivatives then f()g()d = f()d g()d (j) Out of all rectangles with perimeter of 2, the one with the greatest area is a square with side lengths 5. (k) Newton s method may fail to converge if the wrong value is chosen. 2. Let f() = (a) Eplain why f() has a root in the interval [2, 3]. (b) Use Newton s method with = 2 to calculate the root in this interval up to 3 decimal places. 3. Find the general solution of the differential equation y = y 2. You do not need to solve for y in your answer. 4. Sketch the graph of a function f() such that: f() is increasing on (, ), (5, ) and decreasing elsewhere. f() has one vertical asymptote at =. f() is concave up on ( 5, ), (, 8) and concave down elsewhere. f() DNE f() =. Make sure to label points of inflection and local etrema. 5. If f () = then find all critical points on the interval [, ]. 6. A farmer is building a rectangular fenced area with one side bordered by a barn. He has 2 feet of fencing. Draw an appropriately labeled picture of the situation and find an equation of one variable for the Area of his fenced region. (Do not find the maimum area). 7. A farmer needs to build a fenced in region with a particular area. His land is an odd shape such that the perimeter of his fence is P = 3 + 2, where is some parameter related to area. Find the minimum perimeter of a fence he can build with the required area. 3

4 CHAPTER 4. True/False. Eplain your answer. (a) The 2nd Fundamental Theorem of Calculus does not apply if f() is not continuous. (b) If F () is an antiderivative of f() then d/d(f ()) = f(). (c) If F () is an antiderivative of f() then f() = F (). (d) The Parabolic Rule for approimating area is always more accurate than the Right Riemann Sum. (e) Where u = 3. (3 2 ) cos( 3 )d = cos(u)du 2. Use the given partition and sample points i to approimate the area under the curve in the given interval: f() = 2 P = 4 < 5 < 6.5 < 9 = 4, 2 = 6, 3 = Set up the Riemann sum used to calculate the following definite integrals (but do not evaluate the sum) d 4. Evaluate the following sums. 5 ( + 3)d n n i= 2 i= n 2i 2 3i + 7 n i= n (2i n )2 n ((i/n) 3(i/n)2 + ) 4

5 5. Calculate G () for the given functions G(). G() = G() = 3 2 sin(t 2 )dt t2 + 3dt G() = (3t + 2) 4 dt 6. Use the given information to calculate the definite integrals. Given: Find: f()d = 5, f()d = 3 3f()d 2f()d 7. Evaluate the definite integrals using the 2nd Fundamental Theorem of Calculus π/2 4 5 d cos d d 8. Calculate the following indefinite integrals using u-substitution: ( )d sin d 5

6 4 tdt d 9. Evalute each definite integral by using the given u to perform u-substitution: 2 ( ) 5 d, u = ( ) Hint: if u = then = u + d, u = π/2 π cos( 2 )d, u = 2 cos sin(sin )d, u = sin. Eplain why the following integral is equal to without calculating it directly: π/2 π/2 2 sin + 6 d =. Do the following functions achieve their average value on the given interval? Why or why not? f() = 2 + 3, [4, 8] g() = /, [, ] 2. Write the epression (but do not calculate) that would be used to approimate the area under f() = 2 from to 4 using 4 intervals of equal length with: (a) The Right Riemann Sum (b) The Parabolic Rule (For eample, your answer for the Trapezoid Rule would be 4 8 (2 + 2( 2 ) + 2(2 2 ) + 2(3 2 ) )) 6

7 CHAPTER 5. Find the area between the curves from = to = 4: (a) f() = 2 + 3, g() = (b) f() = 2, g() = 4 2. Find the volume of a cone of height and radius by the following steps: STEP : Draw a triangle in the plane bounded by the lines y =, y =, and =. STEP 2: Represent the volume of the cone by a definite integral from = to = with circular cross sections, the circle diameter given by the two functions above. STEP 3: Evaluate the definite integral. 7