Rasmussen College - MH 200 - Week 4 Assignment Problem 1 1. Assume that the number of homicide in Chicago can be modeled by the function R(t) below, where t is the tim R'(4) represent? ( )=1.57 ^2+119 +125 Type your answer here. ) below, where t is the time since year 2000. What does Rasmussen College - MH 200 - Week 4 Assignment Problem 2 2. Assume that the rate of homicide in Chicago can be modeled by the function R(t) below, where t is the time sin R'(6) and explain the difference. ()=1.57^2+119+125 ( 7 9 Type your answer here. below, where t is the time since year 2000. Compute the values of R'(4) and Rasmussen College - MH 200 - Week 4 Assignment Problem 3 3. If you drop a rock off a bridge and its distance is given by H(t) in ft where t is in seconds, then what is the veloc (()=16^2 6 Type your answer here. nds, then what is the velocity of the rock after 2 seconds? Rasmussen College - MH 200 - Week 4 Assignment Problem 4 4. Find the derivative using implicit differentiation. 2 +3 +15=0 Type your answer here. Rasmussen College - MH 200 - Week 4 Assignment Problem 5 5. Assume that C(x) represents the cost of producing x calculators. If C(100) = $500 and C'(100) = $4.75, what is th when producing 100 calculators? Type your answer here. C'(100) = $4.75, what is the average cost of a calculator Rasmussen College - MH200 - Midterm Problem 1 1. Given a differentiable function, what are the relationships among the slope of the tangent line, the instantaneo of the derivative at a point? Type your answer here. ngent line, the instantaneous rate of change and the value Rasmussen College - MH200 - Midterm Problem 2 2. The second derivative of function f(x) is sometimes written as f''(x). Find f''(x) given the function below. ( )=3 ^3+5 ^2+6 Type your answer here. n the function below. Rasmussen College - MH200 - Midterm Problem 3 3. Find the derivative of the function below. ()=^2 ( = ( 2 (+4)2/(^2+1) ( Type your answer here. Rasmussen College - MH200 - Midterm Problem 4 4. Find dy/dx of the function below. =sin +(4 cos )/ Type your answer here. Rasmussen College - MH200 - Midterm Problem 5 5. The function below gives the position of an object in feet moving horizontally after t seconds. Determine a) the of the object at t = 1. ( ( 2 )=2 1 ^32 0 1 ^2+60 ;0 6 Type your answer here. a) b) seconds. Determine a) the velocity and b) the acceleration Rasmussen College - MH200 - Midterm Problem 6 6. Find the slope of the equation below at the given point. =4 /( ^2+3),(3,1) Type your answer here. Rasmussen College - MH 200 - Week 5 Assignment Problem 1 1. Find the critical points of the function below. ( )= ^3+3 ^2+1 Type your answer here. Rasmussen College - MH 200 - Week 5 Assignment Problem 2 2. Find and classify (categorize the point as maximum, minimum or inflection point) of the function below. ()= ( ^33^2+1 3 Type your answer here. of the function below. Rasmussen College - MH 200 - Week 5 Assignment Problem 3 3. Find and classify (categorize the point as maximum, minimum or inflection point) of the function below. ()=^5 ( = Type your answer here. the function below. Rasmussen College - MH 200 - Week 5 Assignment Problem 4 4. Determine the concavity and relative extrema of the function below. ( ( = )=2^432 +56 Type your answer here. Rasmussen College - MH 200 - Week 5 Assignment Problem 5 5. Assume that f(x) below shows the profit in dollars when x units of goods are produced. What are the 1) maxim and 2) the maximum profit? ()=2/3 ^3+9.5^2+10 , 0 Type your answer here. ed. What are the 1) maximum units of production possible Rasmussen College - MH 200 - Week 6 Assignment Problem 1 1. If you were to grow plants in a rectangular-shaped garden with a perimeter of 10 ft, what is the maximum area Type your answer here. what is the maximum area you can achieve? Rasmussen College - MH 200 - Week 6 Assignment Problem 2 2. Assume that the function H(t) gives the height in ft of a stone thrown vertically upward at a velocity of 20 ft/s. ( )=4.9 ^2+20 +2 Type your answer here. ward at a velocity of 20 ft/s. How high does the stone go? Rasmussen College - MH 200 - Week 6 Assignment Problem 3 3. Back at work as a clown, you are pumping gas into a spherical balloon so that its volume increases at a rate of change of its radius when it is 6 inches? Type your answer here. ume increases at a rate of 2 in3/s. What is the rate of Rasmussen College - MH 200 - Week 6 Assignment Problem 4 4. Assume that ABC Airlines has a policy that states the baggage must be box-shaped and its sum of height, width inches. 1) What dimensions (length, width, height) would give the maximum volume and 2) what is the maxim Type your answer here. nd its sum of height, width and length must not exceed 108 and 2) what is the maximum volume? Rasmussen College - MH 200 - Week 6 Assignment Problem 5 5. Given that a company's net sales in billion is given by S(t) where t is the number of years since 2000. What wer change in sales in year 2010? ( ( 5 )=0.2 3 0 5 ^3+2.3^2+10 Type your answer here. 0 ears since 2000. What were the 1) net sales and 2) rate of Rasmussen College - MH200 - Midterm Problem 1 1. Given a differentiable function, what are the relationships among the slope of the tangent line, the instantaneo of the derivative at a point? 1)Let the given function be y f x Then slope of tangent line is y dy x f ' x x dx dy f ' x and instannaeous rate of change is dx ngent line, the instantaneous rate of change and the value Rasmussen College - MH200 - Midterm Problem 2 2. The second derivative of function f(x) is sometimes written as f''(x). Find f''(x) given the function below. ( )=3 ^3+5 ^2+6 f'(x)=3(3x^2)+5(2x)+6(1) =9x^2+10x+6 f''(x)=9(2x)+10(1) =18x+10 n the function below. Rasmussen College - MH200 - Midterm Problem 3 3. Find the derivative of the function below. ()=^2 ( = ( 2 (+4)2/(^2+1) ( 2x 2x x3 4 x 2 2 x 1 x 1 2 x 1 2 2 x 2 x 2 2x2 2 2 f ' x 3 x 8 x 3 x 8 x 2 2 2 x 1 1 x2 2) f x x 2 x 4 2 Rasmussen College - MH200 - Midterm Problem 4 4. Find dy/dx of the function below. =sin +(4 cos )/ 4cos x 4) y sin x x x sin x cos x 1 4 cos x x sin x dy 4cos x 4cos x cos x 1 cos x 1 4 2 2 dx x x x x n x 4cos x cos x x Rasmussen College - MH200 - Midterm Problem 5 5. The function below gives the position of an object in feet moving horizontally after t seconds. Determine a) the of the object at t = 1. ( ( 2 )=2 1 ^32 0 1 ^2+60 ;0 6 Type your answer here. a) velocity v(t)= f'(t) =6t^2-42t+60 so v(1) = 6-42+60=24 ft/sec b) acceleration a(t) = f''(t) = 12t-42 so a(1) = -30 ft/sec^2 seconds. Determine a) the velocity and b) the acceleration Rasmussen College - MH200 - Midterm Problem 6 6. Find the slope of the equation below at the given point. =4 /( ^2+3),(3,1) 2 dy x 3 4 4 x 2 x 12 4 x 2 slope is given by 2 2 dx x 2 3 x 2 3 2 12 4 3 dy 24 1 So slope at point 3,1 is 3,1 2 2 dx 3 3 144 6 Rasmussen College - MH200 - Midterm Problem 1 1. Given a differentiable function, what are the relationships among the slope of the tangent line, the instantaneo of the derivative at a point? 1)Let the given function be y f x Then slope of tangent line is y dy x f ' x x dx dy f ' x and instannaeous rate of change is dx ngent line, the instantaneous rate of change and the value Rasmussen College - MH200 - Midterm Problem 2 2. The second derivative of function f(x) is sometimes written as f''(x). Find f''(x) given the function below. ( )=3 ^3+5 ^2+6 f'(x)=3(3x^2)+5(2x)+6(1) =9x^2+10x+6 f''(x)=9(2x)+10(1) =18x+10 n the function below. Rasmussen College - MH200 - Midterm Problem 3 3. Find the derivative of the function below. ()=^2 ( = ( 2 (+4)2/(^2+1) ( 2x 2x x3 4 x 2 2 x 1 x 1 2 x 1 2 2 x 2 x 2 2x2 2 2 f ' x 3 x 8 x 3 x 8 x 2 2 2 x 1 1 x2 2) f x x 2 x 4 2 Rasmussen College - MH200 - Midterm Problem 4 4. Find dy/dx of the function below. =sin +(4 cos )/ 4cos x 4) y sin x x x sin x cos x 1 4 cos x x sin x dy 4cos x 4cos x cos x 1 cos x 1 4 2 2 dx x x x x n x 4cos x cos x x Rasmussen College - MH200 - Midterm Problem 5 5. The function below gives the position of an object in feet moving horizontally after t seconds. Determine a) the of the object at t = 1. ( ( 2 )=2 1 ^32 0 1 ^2+60 ;0 6 Type your answer here. a) velocity v(t)= f'(t) =6t^2-42t+60 so v(1) = 6-42+60=24 ft/sec b) acceleration a(t) = f''(t) = 12t-42 so a(1) = -30 ft/sec^2 seconds. Determine a) the velocity and b) the acceleration Rasmussen College - MH200 - Midterm Problem 6 6. Find the slope of the equation below at the given point. =4 /( ^2+3),(3,1) 2 dy x 3 4 4 x 2 x 12 4 x 2 slope is given by 2 2 dx x 2 3 x 2 3 2 12 4 3 dy 24 1 So slope at point 3,1 is 3,1 2 2 dx 3 3 144 6 Rasmussen College - MH 200 - Week 4 Assignment Problem 1 1. Assume that the number of homicide in Chicago can be modeled by the function R(t) below, where t is the tim R'(4) represent? ( )=1.57 ^2+119 +125 R'(t) = -3.14t^2+119 R'(4)=68.76 This indicate that rate of change in homicide is 68.76 values per year in 2004 ) below, where t is the time since year 2000. What does Rasmussen College - MH 200 - Week 4 Assignment Problem 2 2. Assume that the rate of homicide in Chicago can be modeled by the function R(t) below, where t is the time sin R'(6) and explain the difference. ()=1.57^2+119+125 ( 7 9 R'(t) = -3.14t^2+119 R'(4)=68.76 R'(6)=5.96 the difference R'(6)-R'(4)= -62.8 homicide per year This indicate that the rate of change in rate of homicide is decrasing by 62.8 values per year from 2004 to 2006. below, where t is the time since year 2000. Compute the values of R'(4) and Rasmussen College - MH 200 - Week 4 Assignment Problem 3 3. If you drop a rock off a bridge and its distance is given by H(t) in ft where t is in seconds, then what is the veloc (()=16^2 6 velocity is v(t)=h'(t)=32*t so velocity after 2 seconds is v(2)=32(2)= 64 ft/sec nds, then what is the velocity of the rock after 2 seconds? Rasmussen College - MH 200 - Week 4 Assignment Problem 4 4. Find the derivative using implicit differentiation. 2 +3 +15=0 differentiating with respect to x we get, 2dy/dx+3+0=0 =>dy/dx= -3/2 Rasmussen College - MH 200 - Week 4 Assignment Problem 5 5. Assume that C(x) represents the cost of producing x calculators. If C(100) = $500 and C'(100) = $4.75, what is th when producing 100 calculators? average cost Avg(x)=C(x)/x So Avg(100)=C(100)/100=500/100=$5 per calculator C'(100) = $4.75, what is the average cost of a calculator Rasmussen College - MH 200 - Week 5 Assignment Problem 1 1. Find the critical points of the function below. ( )= ^3+3 ^2+1 Type your answer here. Rasmussen College - MH 200 - Week 5 Assignment Problem 2 2. Find and classify (categorize the point as maximum, minimum or inflection point) of the function below. ()= ( ^33^2+1 3 Here the critical points are x=0,-2 the point (0,1) is neither maxima nor minima (-2,-3) is a point of minima (-1,-1) is a inflection point of the function below. Rasmussen College - MH 200 - Week 5 Assignment Problem 3 3. Find and classify (categorize the point as maximum, minimum or inflection point) of the function below. ()=^5 ( = here the critical point is x=0 but the point (0,0) is neither maxima nor minima So no global maxima or minima exist (0,0) is a inflection point the function below. Rasmussen College - MH 200 - Week 5 Assignment Problem 4 4. Determine the concavity and relative extrema of the function below. ( ( = )=2^432 +56 the critical point is (2,8) The point (2,8) is a point of minima No global maxima exist since f''(x)=12x^2>0 for all x so the function is concave up in interval (-infinity, infinity) The function is not concave down Rasmussen College - MH 200 - Week 5 Assignment Problem 5 5. Assume that f(x) below shows the profit in dollars when x units of goods are produced. What are the 1) maxim and 2) the maximum profit? ()=2/3 ^3+9.5^2+10 , 0 1) to find maximum units of production maximize f(x) with respect to x. Here for x=10 the function f(x) is maximized Hence maximum unit of production is 10 units 2) maximum profit f(10)= 383.33 dollars ed. What are the 1) maximum units of production possible Rasmussen College - MH 200 - Week 6 Assignment Problem 1 1. If you were to grow plants in a rectangular-shaped garden with a perimeter of 10 ft, what is the maximum area Here let x denotes length and y width of garden Then we mximize area function A= x(5-2x) with respect to x. For x=2.5 the area is maximized Hence length is 2.5 ft and width is (10-2(2.5))/2=2.5 ft what is the maximum area you can achieve? Rasmussen College - MH 200 - Week 6 Assignment Problem 2 2. Assume that the function H(t) gives the height in ft of a stone thrown vertically upward at a velocity of 20 ft/s. ( )=4.9 ^2+20 +2 To find the height the stone goes maximize h(t) with respect to t Here h(t) is maximized when t=20/9.8 sec so maximum height is h= 22.4081 ft ward at a velocity of 20 ft/s. How high does the stone go? Rasmussen College - MH 200 - Week 6 Assignment Problem 3 3. Back at work as a clown, you are pumping gas into a spherical balloon so that its volume increases at a rate of change of its radius when it is 6 inches? rate of change in radius = (1/4*pi*r^2)*dV/dt ~ 0.0044 in/sec ume increases at a rate of 2 in3/s. What is the rate of Rasmussen College - MH 200 - Week 6 Assignment Problem 4 4. Assume that ABC Airlines has a policy that states the baggage must be box-shaped and its sum of height, width inches. 1) What dimensions (length, width, height) would give the maximum volume and 2) what is the maxim 1) For this we maximize V=w^2(108-2w) for w. So we get w=36 inches it maximize volume Hence height =36 inches, width =36 inches, length = 36 inches 2) maximum volume is 46656 inch^3 nd its sum of height, width and length must not exceed 108 and 2) what is the maximum volume? Rasmussen College - MH 200 - Week 6 Assignment Problem 5 5. Given that a company's net sales in billion is given by S(t) where t is the number of years since 2000. What wer change in sales in year 2010? ( ( 5 )=0.2 3 0 5 ^3+2.3^2+10 0 1) net sales in 2010 are s(10)=-0.25(10^3)+2.3(10^2)+10(10)=80 billion 2) rate of change s'(t)=-0.75x^2+4.6x so in 2010 is s'(10)=-29 billion per year ears since 2000. What were the 1) net sales and 2) rate of