Calculus Questions

approximately what values of x between 5 and 5 does the original function y have inflection points W M 3 OA x 2 03 x 0 x 2 03 OB x 2 03 x 2 03 OC x 3 83 x 3 83 OD x 4 91 x 2 03 x 0 x 2 03 x 4 91 O E There are no points of inflection

Here are four graphs Which of the following is the graph of an odd function Graph I Graph II Graph III Graph IV OA Graph II only OB Graph III only OC Graphs III and IV only D Graphs I II and III

1 Consider the family of curves described by x xy ay b Remember to justify your answers A Let a 1 and b 7 Find the two x intercepts B Let a 1 and b 7 Using the two points you found in part A show that the tangents to the curve at these points are parallel and find the common slope D

Use the Runge Kutta implementation in Matlab ode45 to find a numerical solution y 0 6 to the differential equation dy t y t 3 sin 4t y t t y 0 0 dt Give your answer correct to 3 decimal places Number Note the Matlab in the lecture notes produces a plot of the solution to get a numerical answer just leave out the plot command and leave out the semicolon at the end of the ode command then the numerical values will be output to the screen The last one is the result of the last step

Let f x x 4 Which of these is the equation for the normal line to this curve at the point 2 4 O A y 1 2x 25 B y 12x 4 c y 1 2x 5 6 25 D y 12x 6 O E y x 2 12 4 27

1 For each of the following find lim f x x a I II III lim f x x a lim f x x a A f x 2 x 3x 2 c f x 1 2 x 4 sinx if x B f x tanx if x cosx if x X 3 x 3 2 s n 2 for a 2 T TC TT if x if x T a TO a

2 A convex lens with focal length fcentimeters will project the image of an object on a point behind the lens If an object is placed a distance of p centimeters from the lens then the distance q centimeters of the image from the lens is related to p and f by the lens equation 14 1 pq f object K p image A If the focal length of the convex lens is supposed to be 5 cm and if the image is formed 7 cm from the lens find the distance from the lens to the object p It s not necessary to simplify your answer B Find an expression that gives q as a function of p assuming that the focal length is a constant of 5 centimeters C Sketch a graph of q as a function of p i e q p assuming that the focal length is a constant of 5 centimeters Show any important features of the graph

2 Suppose a coin is dropped from the top of the Empire State building in New York which is 1 454 feet tall The position function for free falling objects is s t 16t vot Sor where vo is the initial velocity and so is the initial position A Determine the position and velocity functions for the coin B Determine the average velocity of the coin on the interval 1 3

The sides of the triangle shown increase in such a way that dz dx dy 1 and 3 At the instant when x 12 and y 5 what is the value dt dt dt of y dx dt X O A 39 41 OB 27 O C 13 OD 13 OE 13 E 17 27 13 41 N

Use the notion of signed area to evaluate the definite integral 2 x dx PROTIGHE 6 You get fo 2 x dx OA 8 OB 4 O C 12 OD 6 OE 4

Consider the function f x The graph is below f Use the notion of signed area to evaluate the following definite integral f x dx OA 4 TT OB 2 2T OC 4 4T OD 2 4TT OE 4 2TT 4 4 if 2 x 2 x 2 if 2 x 4

The following graph of f x consists of line segments and semicircles y f x M N 0 13 Click here for long description Use the graph and the notion of signed area to evaluate the definite integral fi f x dx You get OA 18 4x OB 14 x OC 16 2x OD 14 4x 7 8 9 10 11 12 E 16 2 5x

You re given that 1e dx 51 880 to three decimal places Using properties of definite integrals and exponentials find 2e dx A 224 273 OB 143 931 C Can t tell from the information given OD 2084 075 OE 224 273

Given that f3 dx 63 0 and f3 xdx OB 99 0 OC 324 0 OD 576 A Can t tell from the information given E 117 0 13 5 then 3 2x 3 dx

Answer the following questions about the sphere whose equation is given by x y 2 8x 4y 80 1 Find the radius of the sphere Radius r O 2 Find the center of the sphere Write the center as a point a b c where a b and care numbers

Find the equation of the sphere of radius 6 centered at 1 5 1 This sphere is given by

What is the terminal point of the vector a Answer 7 A a 1 4 based at P 2 1

Find a unit vector u in the direction opposite of 3 9 10 u

Find the length of OR R 6 1 9 OR C

and Dot Products Consider the vectors u 4 0 10 and v 2 1 9 Compute u 2 1 9 Compute v 2 29 Compute u v 86

P 4 5 8 R 9 3 8 Find the point Q such that R is the midpoint of Po Q

v 1 w 3 The angle between v and w is 1 3 radians Given this information calculate the following a v w b 4v 2w c 3v1w

Distance and Dot Products Consider the vectors Compute u Compute v Compute u v u 4 0 10 and v 2 1 9

Use the geometric definition of the cross product and the properties of the cross product to make the following calculations a k 3 K b i j ix 3 X c 2kx k 3 d i 3 3 LED

Find the unit vector in the direction opposite to v 4 4

Find the components and length of the following vectors i 5j Components Length i 5j Components Length 4i 5j Components Length 4i 2j Components Length A B 1 0 2 10 4 P

Calculate vx w v 1 8 5 w VX W 4 7 1 W

Calculate the cross product assuming that u x w 5 2 7 3u 3w x w CURS

v 1 6 2 w 8 10 1 Find the cosine of the angle between v and w cos 8

Find a unit vector e where v 9 8 9 ev es

Assume that u v 4 u 5 and v 3 What is the value of 9u u 7v

Consider a triangle ABC like the one below Suppose that A 25 C 117 and a 39 The figure is not drawn to scale Solve the triangle Round your answers to the nearest tenth If there is more than one solution use the button labeled or B 0 b 0 or X No solution S

Rationalize the denominator and simplify 7 2 7 2 LE

The regions in the graph indicate whether a person is obese overweight borderline overweight normal weight or underweight The formula to find a person s body mass index BMI is as follows BMI 703W H The variable W represents weight in pounds The variable H represents height in inches 2 Body Mass Index BMI C 15 Obise 10 20 30 Males Overweigh Borderline Overv Normal Range Underweight 40 50 a Using the formula for BMI find a male s BMI if he is 20 years old 74 inches tall and weighs 234 pounds Round to the nearest integer as needed 60 70

1 point Find the cross product i e j e k 6i e j e k

Let a 5 3 9 and b 9 7 8 be vectors Compute the cross product a x b 7 axb

Find a non zero vector a perpendicular to the vectors V 7 3 and u

Use the graph to find lim f x 10 8 6 4 lim f x X At 2 104 8 6 4 2 0 2 4 6 8 10 f x 2 4 6 O 8 10

of the definite integral we have 6k 6km From the definition limn 0 1 sin k 1 n 6 A fo 2x sin 2 x dx 6 O B f x sin x dx 3 O c f x sin 2x dx O D Jo 2x sin 2x dx 6 O E f x sin 2 x dx

Write the expression as a single logarithm W log3y 4log3z log3 w 3 log

Here is a graph of the function g 54 3 2 m s 40 W ea 1 H 2 3 4 2 Use the graph to find the following If there is more than one answer separate them with commas a All local minimum values of g b All values at which g has a local minimum

Graph the solution to the inequality on the number line w 4 7

Evaluate the following definite integral to three decimal places by 2 interpreting it in terms of areas 9 x 1 2 dx 1 OA 7 0686 OB 34 274 O C 13 069 OD 20 137 OE 8 3562

Factor the following expression 2 3 28475 16xx

Find the exact value of cos x under the given conditions 15 17 0 B sin OA OB 5 13 O D T 0 x cos 2 220 221 140 221 171 OC 221 21 221 Question 7 of 12

Use the information given about the angle 0 0 0 2x to find the exact value of cos 20 3 cos8 7 csc8 0 OA OB O c O D 12 10 49 31 49 12 10 49 Question 8 of 12 31 49

A 1 0 L buffer solution is 0 060 M HC2H302 and 0 250 M LIC2H302 Which of the followi actions will destroy the buffer O adding 0 060 moles of NaOH O All of the above will destroy the buffer O adding 0 060 moles of HC2H302 O adding 0 060 moles of LiC2H302 O adding 0 060 moles of HCI

5 Let g be the function given by g x x 4x 6x 4x k A On what intervals is g increasing Justify your answer I where B On what intervals is g concave upward Justify your answer IS d

hours The cost for each firefighter is 20 for transportation plus 25 per hour for salary meals and other overhead A Let C be the cost of the fire including the cost of the firefighters and the lost timber C will depend on both h how many hours the fire burns and x the number of firefighters sent to fight it Give a formula for C in terms of x and h I

3 Below is the graph of the velocity of an object moving in the positive direction along the x axis Answer the following questions using only information from the graph A 2 3 4 6 t 11 11 21 7 A Find a time t when the object changes directions If this never happens say This never happens Explain your answer B Find the time when the object is furthest to the right Explain your answer I C What is the object s maximum acceleration Explain how you found this