MAC 2311 Final Exam Review Fall Private-Appointment, one-on-one tutoring at Broward Hall

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1 Fall 2016 This review, produced by the CLAS Teaching Center, contains a collection of questions which are representative of the type you may encounter on the eam. Other resources made available by the Teaching Center include: Walk-In tutoring at Broward Hall Private-Appointment, one-on-one tutoring at Broward Hall Walk-In tutoring in LIT 215 Supplemental Instruction Video resources for Math and Science classes at UF Written eam reviews and copies of previous eams The teaching center is located in the basement of Broward Hall: You can learn more about the services offered by the teaching center by visiting

2 1. For each function-interval pair below, find all roots, etrema, and concavity information. Sketch a graph, labeling local and absolute etrema, inflection points, and any other features of interest. (a) f() = on the interval [ 1, 3] y g() = on the interval [ 5, 5] y CLAS Teaching Center 2

3 2. Let f() = 3. (a) Find the linearization of f at = 27. Use the result from part (a) to approimate the cube-root of (c) Use the result from part (a) to approimate the cube-root of 30. (d) Which of these do we epect to be more accurate and why? 3. Find two nonnegative numbers whose sum is 9 such that the product of one number with the square of the other is maimum. 4. Given y = y3/ (3 + 2) 5, calculate dy d. 5. The radius of a sphere is increasing at a rate of 4 millimeters per second. How fast is the volume of the sphere increasing when its diameter is 80 millimeters 6. Find the point on the graph of y = 4 3/2 which is nearest to (2, 4). 7. Calculate the following limits: cos 2 (2) (a) lim 3 2 tan(2) lim 0 (c) lim 0 (d) lim sin(π) (e) lim ( ) 8. Determine, if possible, a value for A such that the piecewise function, f(), below is continuous. If no such number eists, eplain why. sin(π) 0 f() = A = 0 CLAS Teaching Center 3

4 9. Epress 2 0 sin() d as a limit of a Riemann (pronounced REE-MAHN) sum. 10. Evaluate the limit of the Riemann Sum eplicitly by epressing each as a definite integral. { n ( ) ( ) } 2 i i 1 (a) lim 3 6 n n n n i=1 n ( ) iπ π lim sin n n n i=1 11. Evaluate the definite integrals below. (a) 4 1 π π d 3/2 ( cos() sin() ) d 12. Suppose f() is a non-negative function having absolute ma M and absolute min m on the interval [a, b]. (a) What is the largest possible value of What is the smallest possible value of b a b a f() d? f() d? (c) Prove your answers for parts (a) and using a picture. CLAS Teaching Center 4

5 13. Use the fundamental theorem of Calculus the evaluate the epressions below d (a) e t2 dt d d dz 5 1+z 2 e ( ln() + 1 ) d 14. Find the area bounded by f() = 3 and the -ais on the interval [ 1, 1]. Sketch the region to see if your answer makes sense. 15. At time t = 0 a bolt falls from a helicopter which is hovering at an altitude of 4096 feet. Due to gravity, the bolt eperiences a constant acceleration towards the earth, a(t) = 32 ft./s 2. (a) Assuming the bolt was at rest when it began to fall, construct its velocity function, v(t). [Hint, the bolt had no initial velocity] Construct the bolt s displacement function s(t). [Hint, the bolt s initial displacement is the altitude of the helicopter] (c) After how long will the bolt hit the ground? 16. Evaluate the following integrals. Some will benefit from a substitution, others will not. (a) (c) sin(2) sin() d 3π/2 0 sin() d cos θ sin 5/2 θ dθ (d) (e) (f) 1 e z e z + z dz ( ) d e 1 ln d 17. Find the area between the given curves f and g on the given intervals (a) f() = and g() = 3 on the interval [0, 1] f() = and g() = 3 on the interval [0, 2] (c) f() = 2 and g() = on the interval [0, 4]. CLAS Teaching Center 5