Calculus Questions

1 The shaded regions R and R shown below are enclosed by the graphs of f x x and g x 2 R R I A Find the x and y coordinates of the three points of intersection of the graphs of f and g B Estimate the area of R to three reasonable places past the decimal C Find the EXACT area of R No calculator approximations

2 The region R is a rectangle with vertices P a Ina Q a 0 S 3 0 and T 3 Ina where 1 a 3 201 1 54 05 1 0 P T 10 15 20 25 30 0 S A Write an expression that gives the area of the rectangle as a function of a B The area of the rectangle is maximized for some c between 1 and 3 Write the expression you would need to solve in order to find c You don t need to find c C What is the rate of change of the area when a 3 if a is decreasing at a rate of da 0 5 units sec You don t need to simplify of course Use appropriate units dt

X O A B M dx 23 2 C 2 23 2 C C 223 23 2 C O D In 23 23 2 C O E x In 23 23 2 C

The region in the first quadrant enclosed by the curves y 0 x 2 x 5and is rotated about the x axis The volume of the solid generated is 1 y 1 x O A In 5 O B T In O C T ln 2 O D In 10 OE E ln

O A e 3 C Fe O B e C OC 255 22 C O D e 3 C O E Fe C

g x x3 2x 1 What is the slope of the tangent line to the curve g x at the point 2 1 O A 1 OB 5 O C 8 OD 14 O E

The position of an object moving vertically along a line is given by the function s t 4 9t2 26t 17 Find the average velocity of the object over the following intervals a 0 7 b 0 4 c 0 1 d 0 h where h 0 is a real number

A 25 ft ladder is leaning against a vertical wall At what rate with respect to time is the angle between the ground and the ladder changing if the top of the ladder is sliding down the wall at the rate of rinches per second at the moment that the top of the ladder is h feet from the ground You re looking here for an equation in terms of h and 0 A 25 1 4 O A OB O C E 25 1 13 300 A 25 ft 12 1 4 1 25 1 4

3 a From the given graph of f state the numbers at which f is discontinuous and explain why b For each of the numbers stated in part a determine whether f is continuous from the right or from the left or neither 4 2 YA 0 2 4 6

O A 2 ln sin 0 2 4 C OB In sin 0 2 4 C O c 2 ln cos 0 2 4 C O D In cos 0 2 4 C OE None of these s

g x x 2x 1 What is the slope of the tangent line to the curve g x at the point 2 1 O A 1 OB 5 OC 8 OD 14 OE E

z 2 x y is an equation for the tangent plane to z f x y at the point 0 0 Which of the following is the linear approximation for f 0 02 0 01 O2 03 1 97 1 99 O2 01

Suppose that f x y is a differentiable function Which of the following are correct statements Select all that apply fx x y lim h 0 O O fy x y f x y fy x y lim h 0 f x h y f x y h f x y lim f x y h f x y h f x h y h f x y h

Suppose z 1 2x 3y is an equation for the tangent plane to z f x y at 0 0 Then we also know that select all that apply Of 1 1 0 Ofz 0 0 2 Of 0 0 3 Of 0 0 1 The vector n 2 3 1 is orthogonal to the tangent plane

2 Iff is continuous on what can you say about its graph

25 x Find the points of discontinuity of f For each of them give the number Cat which the discontinuity occurs and the values of both one sided limits Enter INF for oo INF for DNE for does not exist CND for cannot determine If you find more than one point of discontinuity enter them in order from smallest to largest If there are answer boxes left unfilled fill each of them with an x Point 1 C lim f x x C lim f x x CH Point 2 C lim f x I C lim f x x C Point 3 C lim f x I C lim f x

Consider the graph of the function g given by g x Remember that one Let A x be the area of the shaded rectangle shown in the figure below where x is the coordinate of the lower right corner of the rectangle with x 0 0 81 0 6 0 4 0 2 20 40 80 Click here for long description The x value x 0 that maximizes A x is 0 OA 1 9802913 B 19 7540002 O C 17 6330301 D 25 9392111 60 100 In x 1 C OE None of the above there is no maximum for 0 x

Hint try the substitution u x O A arcsin x C OB B arcsin u C c arcsin x C O C D x arcsin x C O E 2 arcsin z C

lim f x Find the indicated one sided limits of f and determine the continuity of f at the indicated point NOTE Type DNE if a limit does not exist You should also sketch a graph of y f x including hollow and solid circles in the appropriate places I 6 lim f x I 6 f x lim f x I 6 6 Is f continuous at x 6 YES NO 4 x x 6 5 x x 6

THE RE HERE REFER MEREADERE WASSERANSE ABDGECACE CHARAC 2 If f is continuous on what can you say about its graph Er ARAPRAKINIM

dx O A OB 1 12x O D O E 1 x C None of the above 12 12x 4 1 12x 4

da In 12x 4 dx 1 12x O A OB OC None of the above O D 12 12x 4 OE X 1 12x 4

Let y In t tan a OA 2 tan x sec x O C B 2 tanx sec x tan x secr tan a O D OE Then y tan x x sec x a tan x sec x x tan x sec tan r

The curve y x log3 x 5 has points of inflection at x OA In 3 5 B There are no points of inflection O c 10 O D D 10 In 3 OE 5

da A 2 In 2 In x 2 2 x In 2 In x 2 O B B O C In 2 2 X O D D 2 In 2 2 In x 2 X O E In 2 2 In x 2 X

Let f x 1 x if x 0 2x 1 if x 0 Evaluate f 3 and f 6 f 3 8 f 6 If you have had difficulty with these problems you should look at Sections 1 1 1 3 of y

O A 22 In 2 In x 2 B In 2 In x 2 In 2 2 I OB O C OD In 2 2 ln x 2 O E In 2 2 In x 2

Consider the function f x 20x e on the domain U On its domain the curve y f x A attains its maximum value at x value at x 0 OB attains its maximum value at x minimum value O C attains its maximum value at x value at x 0 OD attains its maximum value at x value OE attains its maximum value at x minimum value and attains its minimum and does not have a and attains its minimum and does have a minimum and does not have a

Find the horizontal and vertical asymptotes for the function f x x 4 x 2 x 3 Vertical asymptotes x 2 x Horizontal asymptote y Enter

What type of function is this 2 f x X 2 1 A odd B neither C even

What type of function is this f x x 1 A neither B odd C even

What type of function is this f x A odd B even C neither

Which of the following is the correct formula for the slope of the tangent line OA Mtan lim OB Mtan lim f t f a t a OD Mtan lim f b f t b t OC Mtan lim t a f t f a t a f b f t b t

What is the slope of the secant line between the points a f a and b f b on the graph of f Which of the following is the correct formula for the slope of the secant line OA Msec OB Msec OC Msec OD Msec f b f a b a f b f a b a f a f b b a f b f a b a

Suppose s t is the position of an object moving along a line at time t 0 What is the average velocity between the times t a and t b Which of the following is the correct formula for average velocity OA Vav OB Vav OC Vav wer Vav s a s b a b s b s a b a s a s b b a s b s a b a

a y c 2 1 0 0 4 1 9 2 15 b d 3x x y 11

3000 young trout are introduced into a large fish pond The number of trout still alive after tyears is modeled by the formula N t 3000 0 9 What is the rate of population decrease when the number of trout in the pond reaches 2000 A Approximately 520 trout per year B Approximately 211 trout per year C Approximately 2000 trout per year D Approximately 494 trout per year OE None of the above

Find intervals where this function is increasing decreasing or constant 3 f x x 3 10 5 1 1 2 3 4 5 6 7 5 10 A increasing from to 3 and decreasing from 3 to o B decreasing from to 3 and decreasing from 3 to o C increasing from t3 and decreasing from 3 to co

Does this function have a discontinuity at x 0 Yes f x x X No

A Find the area of R x B Find the value of h such that the vertical line x h divides the region R into tw regions of equal area C Find the volume of the solid generated when R is revolved about the x axis

For what values does the function fail to exist f x A x 1 4 and x 0 C x 4 x 0 and x 4 X x 16x B x 1 4 x 0 and x 4 D x 1 4 x 0 and x 4

2 River currents miles per hour at a certain location are given below The current direction at this location was from the north during the time interval shown and the current did not exhibit any severe fluctuations other than those shown in the chart Time of Day 6 00 AM 6 10 AM 6 20 AM Speed mph 17 20 22 6 30 AM 21 6 40 AM 17 A Using trapezoids estimate the average river current speed from the north from 6 00 AM until 6 40 AM B At approximately what time between 6 00 AM and 6 40 AM would you estimate that the river current had the average velocity C A message in a bottle near this location is released at 6 00 AM Assuming that the bottle travels along with the river s current approximately how far south will the bottle be at 6 40 AM

For what value does the function fail to exist f x A x 4 C x 4 3 X 4 B x 3 D x 0

The position of a particle moving in a straight line is given by s t et cos 5t for t 0 where tis in seconds If the particle changes direction at time T seconds then Tmust satisfy the equation A 5e t sin 5t 0 OB tan 57 3 OC cos 5T 0 O D 5T arctan 5 E cot 57 5

Find the domain of the function f x x 2 A x22 C no solutions B x 2 D x 2

For what values does the function fail to exist A x 5 and x 0 f x C x 1 5 and x 0 X x2 5x B x 0 and x 5 D x 1 5 and x 0

For this question refer to the following graph and information An ant is crawling on a straight wire The VELOCITY v t of the ant at time 0 t 8 is given in the graph at right Note The graph on 0 t 2 is a semicircle half circle A 4 2 B 5 7 C 8 1 2 D 5 1 2 E 37 5 3 What is the net distance traveled by the ant over the time interval 0 t 8 Click here for long description

da O A O O B O c x 3 x ln 2 1 3x O D None of the above 1 O E x In 2

Find a definite integral for which 1 4 4 k 4 4 4 is a right rectangle approximation 5 O A x dx B x 4x dx c x 4x dx OD 21 6 5 O E x dx

dx O A OB B e3r 1 e z 1 2 2 1 e 1 x 1 2 O c 1 1 2 O D 3 D 2 1 2 E 3 1 x 1 2