Calculus Questions

6200 R v V can be used to determine the heart rate R of a person whose heart pumps 6200 milliliters mL of blood per minute and v milliliters of blood per beat Complete parts a through c below a Find the rate of change of heart rate with respect to v the output per beat R v b Find the heart rate at v 50 mL per beat The heart rate is beats min Round to two decimal places as needed c Find the rate of change at v 50 mL per beat The rate of change is beats min per mL Round to two decimal places as needed Heart rate 200 R v 100 100 Output per beat 200

10t 1 The temperature T of a person during an illness is given by T t a Find the rate of change of the temperature with respect to time b Find the temperature at t 1 hr c Find the rate of change of the temperature at t 1 hr a Find the rate of change of the temperature with respect to time T t b Find the temperature at t 1 hr T 1 F Round to two decimal places as needed c Find the rate of change of the temperature at t 1 hr T 1 F hr Round to two decimal places as needed 98 6 where T is the temperature in degrees Fahrenheit at time t in hours AT 1 104 98 0 10 Q

5 Use the graph and the function to find the following a Find lim k x x 4 b Find k 4 c Is k continuous at x 4 a Select the correct choice below and fill in any answer boxes in your choice OA lim k x x 4 Round to the nearest integer as needed OB The limit does not exist b Select the correct choice below and fill in any answer boxes in your choice OA K 4 Round to the nearest integer as needed OB The function is undefined at x 4 c Is k continuous at x 4 O Yes O No CLB H AN Ay F but d 2 5 4 8 24 K

Find the derivative dx dx 3 X 3 X

List the points in the graph in the interval 1 x 6 at which the function is not differentiable Ay 40 R 8 4 6 8 10 SOU The function is not differentiable at x Use a comma to separate answers as needed

Suppose that a fashion company determines that the cost in dollars of producing x cellphone cases is given by C x 0 05x 60x Find result to the company C 301 C 300 Simplify your answer 301 300 Interpret the significance of this result to the company Choose the correct answer below OA It represents the additional cost to produce one more item after making 300 items B It represents the average cost of producing 301 items C It represents the additional cost to produce one item after the fixed costs have been paid D It represents the average cost per item to produce 300 items C 301 C 300 301 300 and interpret the significance of this

Graph the following function and then find the specified limits When necessary state that the limit does not exist 1 f x 5 find lim f x and lim f x X 818 x 0 OA OA lim f x x48 Find lim f x Select the correct choice below and fill in any answer boxes in your choice 818 OB The limit does not exist B Type an integer or a simplified fraction O A lim f x x 0 Find lim f x Select the correct choice below and fill in any answer boxes in your choice x 0 Q Type an integer or a simplified fraction B The limit does not exist O C Q O D Ay 1

Is the function given by f x 1 x 7 for x 3 4x 4 for x 3 Choose the correct answer below continuous at x 3 Why or why not OA The given function is not continuous at x 3 because f 3 lim f x 8 x 3 OB The given function is continuous at x 3 because f 3 lim f x 8 X 3 OC The given function is not continuous at x 3 because the limit does not exist D The given function is continuous at x 3 because f 3 does not exist

Decide from the graph whether a limit exists If a limit exists find its value lim F x X 4 What is the limit Select the correct choice below and fill in any answer boxes in your choice OA The limit is the real number B The limit does not exist 10 8 B 10 2b 4 B B V FX 10 A Q

Find the derivative of the following function y 9x y 3 2 2x 1 12 8

Is the function g x continuous at x 9 G x 81 No O Yes x 9 19 for x 9 for x 9 Choose the correct answer below

For the polynomial below 1 is a zero 3 f x x 3x 2 Express f x as a product of linear factors f x

Let f x a Use a table of values to evaluate f x 3 X 0 5 0 9 0 99 and consider the limits lim f x and lim f x X 17 x 1 0 999 0 9999 0 99999 lim f x f x 8 X 1 5 1 1 1 01 1 001 1 0001 for values of x that approach 1 from the left and from the right Round your answers to one decimal place 1 1 00001 f x b Use an analytical argument to find each limit If the limit is infinite enter co or 00 as appropriate If the limit does not otherwise exist enter DNE lim f x c Sketch a graph of f to confirm your results A graphing calculator is recommended y y y

Use the graph of g to find the y a b lim g x x 0 1 lim g x e 40 2 c lim g x x 0 DNE d lim g x x 2 2 lim g x X 2 0 f lim g x x 2 g g 2 4 2 20 4 9 x

a f x 16 0 03 0 11 Points Which of the following functions f has a removable discontinuity at a If the discontinuity is removable find a function g that agrees with f for x a and is continuous at a If an answer does not exist enter DNE 1 X 1 g x 1 b f x DETAILS The discontinuity is removable The discontinuity is not removable a 1 x x 6x X 3 g x 4 g x 3 X PREVIOUS ANSWERS SCALCET9 2 5 051 a 3 The discontinuity is removable The discontinuity is not removable X c f x sin x a Recall that The discontinuity is removable The discontinuity is not removable IVIT means the largest integer that is less than or equal to h x

What is the relationship between m and the slopes of the lines of a piecewise function Fon a b Given that F b F a m b a

Question 8 Given f x 1 24 and g z 16 x 3 find the domain of f g x

5 A graphing calculator is recommended If a rock is thrown upward on the planet Mars with a velocity 8 m s its height in meters t seconds later is given by y 8t 1 86t2 Round your answers to two decimal places a Find the average velocity in m s over the given time intervals 1 1 2 1 1 1 5 1 1 1 iv 1 1 01 v 1 1 001 m s m s m s m s m s b Use your answers from part a to estimate the instantaneous velocity of the rock in m s when t 1

Divide 19x 6x 20 1 2x 3 Write your answer in the following form Quotient 19x 6x 20 2x 3 0 X 2x 3 0 2 Remainder x 2x 3

Divide 10x 19x 6x 14 9x 2x 1 Write your answer in the following form Quotient 10x 19x 6x 14 9x 2x 1 0 Remainder 2x 1 0 2x 1

Prove the statement using the definition of a limit 7 2x lim x 1 3 2x en 7 2 3 E Select But Given 0 we need Select such that if 0 x 1 6 then 7 2x Thus lim 3 by the definition of a limit ut 7 2x 3 x x 1 x 1 x 1 Select 2x 7 x So if we choose Select then 0 x 1 8

1 04 Points Sketch the graph of a function that satisfies all of the given conditions lim f x 7 lim f x 5 lim f x 5 f 6 6 f 3 4 X 3 O 5 DETAILS SCALCET9 2 XP 2 006 0 2 Submissions Used Submit Answer y 8 6 X 5 6 5 y 2 5 X 5 61 41 2 5 MY NOTES x

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval e4 3x 0 1 The equation e4 3x is equivalent to the equation f x e 4 3x 0 f x is continuous on the interval 0 1 f 0 Intermediate Value Theorem Thus there is a root of the equation e 4 3x in the interval 0 1 and f 1 Since Select 0 Select there is a number cin 0 1 such that f c

4 1 04 Points Find each x value at which is discontinuous and for each x value determine whether fis continuous from the right or from the left or neither x 2 X DETAILS SCALCET9 2 5 045 0 2 Submissions Used X 10 x 2 ifx 0 Isf continuous from the right left or neither at this value O continuous from the right O continuous from the left Oneither if 0 x 1 if x 1 8 X smaller value f x ex larger value Is f continuous from the right left or neither at this value O continuous from the right O continuous from the left Oneither Sketch the graph of f 5 y 10 5 5 10 5 10 X 10 5 DO y 10 5 5 10 5 10 X 10 DO 5 y 10 5 5 10 5 10 X 10 DO 5 y 10 5 5 10F 5

Find all x intercepts and y intercepts of the graph of the function f x 3x 3x 18x If there is more than one answer separate them with commas Click on None if applicable x intercept s y intercept s None 3 X 0 0

Explain what it means to say that lim f x 8 and X 3 X 3 O As x approaches 3 from the right f x approaches 8 As x approaches 3 from the left f x approaches 5 O As x approaches 3 from the left f x approaches 8 As x approaches 3 from the right f x approaches 5 O As x approaches 3 f x approaches 5 but f 3 8 O As x approaches 3 f x approaches 8 but f 3 5 lim f x 5 In this situation is it possible that lim f x exists Explain X 3 O Yes f x could have a hole at 3 8 and be defined such that f 3 5 O Yes f x could have a hole at 3 5 and be defined such that f 3 8 Yes if f x has a vertical asymptote at x 3 it can be defined such that lim f x 8 lim f x 5 and lim f x exists X 3 X 3 X 3 No lim f x cannot exist if lim f x lim f x X 3

Submit Answer 5 5 X 5 5 DETAILS SCALCET9 2 XP 5 005 0 2 Submissions Used X 5 2 1 04 Points For the function f how would you remove the discontinuity In other words how would you define f 4 in order for f to be continuous at 4 2x 8

Problem 7 Let f x 8 x ln x 7 x 7 a Find 16 b Find 22

Problem 6 Find a formula for the inverse of the following function Set 7e 4 f x A y G y 7et 4et 5 7e 4 Set B y In H y 4x 7x 5 7e 4 Set 5ln x C y7lnx 4 D y 7lnx 4 5in x In Fy In 7 4x

Problem 8 Find the exact value of tan sin

6 7 8 4 S Domain Range Is it a function Domain Range Is it a function Domain Range Is it a function Pag

The dot plot below displays the high temperature F of the day for all 31 days in March 2017 When comparing Baltimore s and San Diego s high temperatures which city has a smaller standard deviation San Diego Baltimore 30 Baltimore San Diego 40 888 8 50 Temperature 60 70 80

The boxplot below displays the Grade Point Averages GPA of a random sample of collegiate athletes Do the male or female athletes have more variability in their GPAS Male Boxplot of GPAS Female H 2 5 Female Male GPA 3 5

A Cepheid variable star is a star whose brightness alternately increases and decreases Suppose that Cephei Joe is a star for which the interval between times of maximum brightness is 4 7 days Its average brightness is 4 4 and the brightness changes by 0 6 Using this data we can construct a mathematical model for the brightness of Cephel Joe at time t where t is measured in days B t 4 4 0 6 sin 2xt 4 7 a Find the rate of change of the brightness after t days Rate of change b Find the rate of increase after one day Rate of increase

Find the 31 th derivative of y cos 32 by finding the first few derivatives and observing the pattern that occurs

A table of values for f g f and g is given below 0 B If H z g f x then H 1 A If h z f g x then h 1 x f x g x x g x 1 3 3 1 1 2 2 2 3 CAD 1 food 2 CAR 1 1 1 1

Suppose that Find an equation for the tangent line to the graph of f at a Tangent line 2 452

Match the functions and their derivatives 1 y cos tan x 2 y sin z tan z 3 y tan z 4 y cos x A y 1 tan z B y sin x tan z sec z cos 1 C y sin tan x cos

Let f x 2x 3 f 3 119

flect Back 2 points In each row of the following table you are given some information about behavior of a function f on an interval a b what if anything can you say about f on the ne interval Pick from the following choices can t say anything positive negative zero increasing creasing constant concave up concave down linear If f x is then f x is positive negative increasing decreasing constant concave up concave down linear

c Sketch the graph of p x arctan x and then sketch the graph of p x List all values of x for which p x is not differentiable 11 loto to b a Explain What does this say about

Given that f x 2x evaluate the function at x 3 6 12 O 6 12 5 1 point If we are evaluating the function f x then we cannot have the following f x 0 when x 0 f x 0 when a 0 All of the other answers are possible f x 0 when x 0

1 2 points Let v 5 2 and w 2 1 Compute and plot on a single set of axes each of the following vectors a v w b v w c proj w d proj 2w V

5 2 point In the graph below a The red line represents all points whose position vectors are orthogonal to a i e all x such that a x 0 y a X Sketch this picture on a set of axes and label all points with position x such that a x 2 and then again for a x 4

3 2 points Let a 1 2 and b 3 4 and c 1 7 be position vectors for the vertices of a triangle Find the average of the vertex positions and use the properties of vector addition and scalar multiplication to show this is the intersection of the medians of the triangle A median of a triangle is a line segment from a vertex to the midpoint of the opposite side

2 2 points Given position vectors v and w which linear combinations cv dw corre spond to the shaded parallelograms below Your answer should be a set of conditions on c and d a W y V X b W y X

4 2 points Show that each statement below is always true or find a counterexample A counterexample should be a concrete set of vectors or scalar for which the statement is false a projab b c compa comp b b proja cu dv cproj u dprojav d a proj a is orthogonal to b

ctor the trinomial 4 a b 2 11 a b 3

Problem 5 Solve the following equation for x In 12ln x 7 Enter your answer symbolically

Problem 4 Simplify the following expressions a e 3 ln 5 In 1 b log2 48 log 3