MAT 270

The rate of change function of a function f is given by rf(x)=x2-2, On a paper define the approximate net accumulation function with Δx=0.3 when 1 ≤ x < 2.2. What is the approximate net change in f from x=1 to x=2?

-0.064 A(2)=(12-2)*0.3+(1.32-2)*0.3+(1.62-2)*0.3+(1.92-2)*0.1

A rocket-powered basketball was launched from the ground. The rocket's velocity was measured, in meter per second, for the first 10 seconds after the launch. The rocket's velocity function v is given by v(t)=t2 , 0 ≤ t ≤ 10. On a paper define the piece-wise defined approximate rate of change function r on the interval [2, 3.2) using the "left end" approximation method with Δt=0.3. Give your answers for each of the 4 values used. r(x)=rate 1, 2 ≤ t < 2.3 r(x)=rate 2, 2.3 ≤ t < 2.6 r(x)=rate 3, 2.6 ≤ t < 2.9 r(x)=rate 4, 2.9 ≤ t < 3.2

1- 2-5.29 3-6.76 4-8.41

The function rv gives the rate of change of the volume of oil expelled per day by a rig that begins leaking at day t=0. The function r gives the approximate rate function with moments of duration 0.3 days for 0≤ t ≤1.5. We use the rates at the left endpoint of each moment to define the constant rate for that moment. Consequently the graph of an approximate rate function is a step function of piece-wise horizontal line segments over each moment of duration Δt=0.3 days. Calculate A(0.7), the approximate amount of oil, in cubic meters, expelled 0.7 days after the leaking started assuming the constant rates: r(t)=r(0)=rv(0)=28 when 0 ≤ t < 0.3, r(t)=r(0.3)=rv(0.3)=20 when 0.3 ≤ t < 0.6 r(t)=r(0.6)=rv(0.6)=12.5 when 0.6 ≤ t < 0.9 r(t)=r(0.9)=rv(0.9)=6 when 0.9 ≤ t < 1.2 r(t)=r(1.2)=rv(1.2)=3 when 1.2 ≤ t <1.5

15.65 28*0.3+20*0.3+12.5*(t-0.6).

The function rv gives the rate of change of the volume of oil expelled per day by a rig that begins leaking at day t=0. The function r gives the approximate rate function with moments of duration 0.3 days for 0≤ t ≤1.5. We use the rates at the left endpoint of each moment to define the constant rate for that moment. Consequently the graph of an approximate rate function is a step function of piece-wise horizontal line segments over each moment of duration Δt=0.3 days. Calculate A(1.1), the approximate amount of oil, in cubic meters, expelled 1.1 day after the leaking started assuming the constant rates: r(t)=r(0)=rv(0)=28 when 0 ≤ t < 0.3, r(t)=r(0.3)=rv(0.3)=20 when 0.3 ≤ t < 0.6 r(t)=r(0.6)=rv(0.6)=12.5 when 0.6 ≤ t < 0.9 r(t)=r(0.9)=rv(0.9)=6 when 0.9 ≤ t < 1.2 r(t)=r(1.2)=rv(1.2)=3 when 1.2 ≤ t <1.5

19.35 28*0.3+20*0.3+12.5*0.3+6*(t-0.9)

The function rv gives the rate of change of the volume of oil expelled per day by a rig that begins leaking at day t=0. The function r gives the approximate rate function with moments of duration 0.3 days for 0≤ t ≤1.5. We use the rates at the left endpoint of each moment to define the constant rate for that moment. Consequently the graph of an approximate rate function is a step function of piece-wise horizontal line segments over each moment of duration Δt=0.3 days. Calculate A(1.4), the approximate amount of oil, in cubic meters, expelled 1.4 days after the leaking started assuming the constant rates: r(t)=r(0)=rv(0)=28 when 0 ≤ t < 0.3, r(t)=r(0.3)=rv(0.3)=20 when 0.3 ≤ t < 0.6 r(t)=r(0.6)=rv(0.6)=12.5 when 0.6 ≤ t < 0.9 r(t)=r(0.9)=rv(0.9)=6 when 0.9 ≤ t < 1.2 r(t)=r(1.2)=rv(1.2)=3 when 1.2 ≤ t <1.5

20.55 28*0.3+20*0.3+12.5*0.3+6*0.3+3(t-1.2)

The following graph represents an approximate net accumulation function A defined on [1, 3) with Δx=1 associated with some rate function. Find the value of the dependent variable of the associated approximate rate of change function at x=1.3. That is, find r(1.3).

4 Since 1 <1.3 <2, r(1.3) is (4-0)/(2-1)=4

Stephanie ran the Boston Marathon in 2014. She finished in 3 hours 51 minutes and 26 seconds. Let the variable y represent the number of miles Stephanie has run. Which of the following is an appropriate definition of the variable dy. A) dy= a variation in the distance, in miles, ran by Stephanie since the start of the race. B)dy= a variation in distance ran by Stephanie since the start of the race. C)dy=a variation in the number of miles. D)dy= a variation in the distance

A) dy= a variation in the distance, in miles, ran by Stephanie since the start of the race.

Check the correct definition(s) of the concept of constant rate of change. y varies at a constant rate m with respect to x exactly when .... A) the ratio of dy and dx is always m, for any dx and corresponding dy, that is dy/dx=m. B)all of these definitions give the comprehensive, equivalent meaning of the constant rate of change C)for a fixed change Δx in x, the corresponding change Δy in y is mΔx D)for a unit change in x, the corresponding change in y is m. E)for any variation, dx, in x is proportional to the corresponding variation, dy, in y, where the constant of proportionality is m. F)for any variation dx in x, the corresponding variation dy in y is m times dx. That is dy=mdx.

A) the ratio of dy and dx is always m, for any dx and corresponding dy, that is dy/dx=m. E)for any variation, dx, in x is proportional to the corresponding variation, dy, in y, where the constant of proportionality is m. F)for any variation dx in x, the corresponding variation dy in y is m times dx. That is dy=mdx.

Consider the following graph displayed below in the xy coordinate plane. Check the statement that are true and only those. A)For every y value the graphs shows exactly one associated value of x. B)The graph shows a relation ship between x and y, where y is a function of x. C)The graph shows a relation ship between x and y, where x is a function of y. D)The graph shows a relation ship between x and y, where x is a function of y and y is a function of x E)For every x value the graphs shows exactly one associated value of y. F)The graph shows a relation ship between x and y, where neither x is a function of y nor y is a function of x

A)For every y value the graphs shows exactly one associated value of x. C)The graph shows a relation ship between x and y, where x is a function of y.

What do you mean by " rate of change at the moment x=a"? A)The constant rate (if it exists) at which the dependent variable changes over a tiny interval of the independent variable containing x=a B)The differential of the independent quantity. C)The rate of change of the dependent variable with respect to exactly one value of the independent variable x=a. D)The differential of the dependent quantity. E) F)

A)The constant rate (if it exists) at which the dependent variable changes over a tiny interval of the independent variable containing x=a

Suppose P=h(t) represents the population (number of people) of the city Phoenix, Arizona, t years after the start of the year 2000. Match the function notation with their meanings. A)h(k-3) B)3h(k) C)h(k)-3 D)h(k)-h(3) E)h(3)-k F)h(k+3)-h(3)

A)The population of Phoenix k-3 years after 2000 B)3 times the population of Phoenix k years after 2000 C)3 people less than the population of Phoenix k years after 2000 D) The variation in the population of Phoenix from 2003 to k years after 2000 E)k people less than the population of phoenix in 2003 F)The variation in the population of Phoenix from 2003 to k+3 years after 2000

In a function f, which of the following notations may represent a way to calculate the new (final) value of the dependent variable resulting from a little change dx in the independent variable from x=a. (Assume the rate of change at the moment x=a exists and it is denoted by m) A)f(a)+mdx B)a+mdx C)mdx D)f(a+mdx)

A)f(a)+mdx

You pull the drain plug from a sink you were washing dishes in. Unfortunately it was partly clogged and the water drains very slowly. Let h denote the height of the water in the sink (in cm) t minutes after pulling the drain plug. Assume water is draining so that its height decreases at a constant rate of 1.23 cm/min. Identify each of the following terms with the best description: A) The letter h which represents the height of the water in the sink (in cm)? B)The time (in minutes) after pulling the drain plug? C)The constant rate 1.23 cm/min at which the water is draining? D)The letter T which represents the total time needed to drain the water from the sink? E) The molar mass of the water which is 1801525 grams/mole?

A)variable B)a varying quantity in this scenario C)a constant quantity in this scenario D) a parameter E) constant

The following graphs show linear relationships between x and y. Use the following graphs and check the statements that are true and only those. A)y/x=2 for each x and corresponding y value only on the graph shown in picture B (but not on picture A) B)dy/dx=2 for any variation, dx, in x and corresponding corresponding variation, dy, in y only on the graph shown in picture B (but not in picture A) C)dy/dx=2 for any variation, dx, in x and corresponding corresponding variation, dy, in yon both graphs shown in picture A and B. D)y=2x+1, for each x and corresponding y value on the graph shown in picture A. E)y/x=2 for each x and corresponding y value only on the graph shown in picture A (but not on picture B) F)dy/dx=2 for any variation, dx, in x and corresponding corresponding variation, dy, in y only on the graph shown in picture A (but not in picture B) G)y/x=2 for each x and corresponding y value on both graphs shown in picture A and B. H)dy=2dx+1 for any variation, dx, in x and corresponding variation, dy, in y on the graph shown in picture A.

A)y/x=2 for each x and corresponding y value only on the graph shown in picture B (but not on picture A) C)dy/dx=2 for any variation, dx, in x and corresponding corresponding variation, dy, in yon both graphs shown in picture A and B. D)y=2x+1, for each x and corresponding y value on the graph shown in picture A.

The function rv gives the rate of change of the volume of oil expelled per day by a rig that begins leaking at day t=0. The function r gives the approximate rate function with moments of duration 0.3 days for 0≤ t ≤1.5. We use the rates at the left endpoint of each moment to define the constant rate for that moment. Consequently the graph of an approximate rate function is a step function of piece-wise horizontal line segments over each moment of duration Δt=0.3 days. Graph the corresponding approximate accumulation function (piece-wise linear) using these 5 rates between t=0 and t=1.5 days and Δx= 0.3. The initial amount of oil spilled is 0. Check the statements that are TRUE and only those. A)A(t)=r(0)·0.3+r(0.3)·0.3+r(0.6)·(t-0.6) when 0.6 ≤ t <0.9 B)A(t)=r(0)·0.3+r(0.3)·(t-0.3) when 0.3 ≤ t <0.6 C)A(t)=r(0)·(t-0) when 0 ≤ t <0.3 D)A(t)=r(0)·0.3+r(0.3)·0.3+r(0.6)·0.3+r(0.9)·0.3 +r(1.2)·(t-1.2) when 1.2 ≤ t <1.5 E)A(t)=r(0)·0.3+r(0.3)·0.3+r(0.6)·0.3+r(t)·(t-0.9) when 0.9 ≤ t <1.2 F)A(t)=r(0)·0.3+r(t)·(t-0.3) when 0.3 ≤ t <0.6 G)A(t)=r(0)·0.3+r(0.3)·0.3+r(0.6)·0.3+r(0.9)·(t-0.9) when 0.9 ≤ t <1.2 H)A(t)=r(0)·0.3+r(0.3)·0.3+r(t)·(t-0.6) when 0.6 ≤ t <0.9 I)A(t)=r(t)·(t-0) when 0 ≤ t <0.3 J)A(t)=r(0)·0.3+r(0.3)·0.3+r(0.6)·0.3+r(0.9)·0.3 +r(t)·(t-1.2) when 1.2 ≤ t <1.5

ALL CORRECT

The graph of y = x 3 displayed below. Select all the statements that are true for this relationship between the variables x and y. The domain for the variable x is the set of real numbers. A)For y=0 there are more than 1 corresponding x values which satisfy the given equation B)For each y value there is exactly one corresponding x value which satisfies the given equation C)For each x value there is exactly one corresponding y value which satisfies the given equation D)x is a function of y but y is not a function of x E)For x=0 there are more than 1 corresponding y values which satisfy the given equation F)x is a function of y and y is a function of x G)Neither x is a function of y nor y is a function of x H)x is not a function of y but y is a function of x I)To decide if there is a function relation between the corresponding x and y values defined by an equation, I use the graph as an aid but mainly I investigate the given equation. If a rule assigns exactly one output for each input, then I know this rule is a function relationship between the inputs and the outputs. J)To decide if there is a function relation between the corresponding x and y values defined by an equation, I only look at the graph and draw a conclusion based on the displayed graph of the equation.

B)For each y value there is exactly one corresponding x value which satisfies the given equation C)For each x value there is exactly one corresponding y value which satisfies the given equation F)x is a function of y and y is a function of x I)To decide if there is a function relation between the corresponding x and y values defined by an equation, I use the graph as an aid but mainly I investigate the given equation. If a rule assigns exactly one output for each input, then I know this rule is a function relationship between the inputs and the outputs.

You pulled the drain plug from a sink you were washing dishes in. The sink was partly clogged and the water drains very slowly. Let h be the height of the water in the sink (in cm) at time t minutes after pulling the plug. Assume the water is draining so that its height decreases at a constant rate of 1.23 cm/min. Identify the letters which represent parameters(s): A)The height of the water (in cm) in the sink since the plug was pulled. B)The letter T, which represents the total time (in minutes) during which the water drained from the sink C)The letter t, which represents the time (in minutes) since the plug was pulled. D)997 kg/m³, which is the density of the water. E)The time (in minutes) since the plug was pulled. F) The letter m, which represents the constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time. G) The constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time H)The letter h, which represents the height of the water (in cm) in the sink since the plug was pulled.

B)The letter T, which represents the total time (in minutes) during which the water drained from the sink F) The letter m, which represents the constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time.

A rocket-powered basketball was launched from the ground. The rocket's velocity was measured, in meter per second, for the first 10 seconds after the launch. The rocket's velocity function v is given by v(t)=t2 , 0 ≤ t ≤ 10. Which of the following expression defines the approximate net accumulation function A in the rocket's height between t=2.6 seconds and t=2.9 seconds ( 2.6 ≤ t < 2.9) assuming we start to measure the accumulation in height at t=2 seconds and we use Δt=0.3 in the approximation. A)A(t)=02*0.3+0.32 *0.3+0.62*(t-0.6) if 2.6 ≤ t < 2.9 B)A(t)=22*0.3+2.32 *0.3+2.62*0.3 if 2.6 ≤ t < 2.9 C)A(t)=22*0.3+2.32 *0.3+2.62*(t-2.6) if 2.6 ≤ t < 2.9 D)A(t)=22*2+2.32 *2.3+2.62*t if 2.6 ≤ t < 2.9 E)A(t)=2.62*(t-2.6) if 2.6 ≤ t < 2.9

C)A(t)=22*0.3+2.32 *0.3+2.62*(t-2.6) if 2.6 ≤ t < 2.9

You pulled the drain plug from a sink you were washing dishes in. The sink was partly clogged and the water drains very slowly. Let h be the height of the water in the sink (in cm) at time t minutes after pulling the plug. Assume the water is draining so that its height decreases at a constant rate of 1.23 cm/min. Identify the letters which represent variable(s): A)The height of the water (in cm) in the sink since the plug was pulled. B)The letter T, which represents the total time (in minutes) during which the water drained from the sink C)The letter t, which represents the time (in minutes) since the plug was pulled. D)997 kg/m³, which is the density of the water. E)The time (in minutes) since the plug was pulled. F) The letter m, which represents the constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time. G) The constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time H)The letter h, which represents the height of the water (in cm) in the sink since the plug was pulled.

C)The letter t, which represents the time (in minutes) since the plug was pulled. H)The letter h, which represents the height of the water (in cm) in the sink since the plug was pulled.

A woman running at a steady speed burns calories at a rate of m = 105 calories per mile. If we were to write "dy = m*dx", what could the dx mean here? A)the total number of calories she burns B)the total number of miles she runs C)an additional number of miles she runs D)an additional number of calories she burns

C)an additional number of miles she runs

If y and x are in a constant rate of change relationship, which of these must be equivalent to dy? (The symbol m is a parameter where seen) A) dy/dx B) m C) y/x D) m*dx E) m*x

D) m*dx

You pulled the drain plug from a sink you were washing dishes in. The sink was partly clogged and the water drains very slowly. Let h be the height of the water in the sink (in cm) at time t minutes after pulling the plug. Assume the water is draining so that its height decreases at a constant rate of 1.23 cm/min. Identify the letters which represent constant(s): A)The height of the water (in cm) in the sink since the plug was pulled. B)The letter T, which represents the total time (in minutes) during which the water drained from the sink C)The letter t, which represents the time (in minutes) since the plug was pulled. D)997 kg/m³, which is the density of the water. E)The time (in minutes) since the plug was pulled. F) The letter m, which represents the constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time. G) The constant rate of change, 1.23 cm/min, at which the height of the water changes with respect to time H)The letter h, which represents the height of the water (in cm) in the sink since the plug was pulled.

D)997 kg/m³, which is the density of the water.

What do you mean by "moment at x=a"? A)It is a small change in time. B)a specific point in time C)It is just dx, other term for differential D)A tiny interval of the independent variable containing x=a

D)A tiny interval of the independent variable containing x=a

The candy mix costs 3 dollars per pound and the box for the candy mix is 1.5 dollars Let the variable x represent the weigh of the candy mix in pounds and the variable y represent the cost of the mix in dollars. Check all the statements that are true and only those. A)For any increase in the weight of the candy mix the corresponding increase in the cost is 1/3 times that increase in the weight. B)If dx represents a variation in x and dy represents the corresponding variation in y, then dy=3dx+1.5 C) For any increase in cost of the candy mix the corresponding increase in the weight is 3 times that increase in the cost. D)If dx represents a variation in x and dy represents the corresponding variation in y, then dy=3dx E)For any increase in the weight of the candy mix the corresponding increase in the cost is 3 times that increase in the weight. F)For any increase in cost of the candy mix the corresponding increase in the weight is 1/3 times that increase in the cost. G) y=3x H) If the weight of the candy mix varies from x=1 to x=7 pounds, then the cost increases by 18 dollars. I)y=3x+1.5 J) The total cost of the candy mix is 3 times as much as the weight of the candy mix.

D)If dx represents a variation in x and dy represents the corresponding variation in y, then dy=3dx E)For any increase in the weight of the candy mix the corresponding increase in the cost is 3 times that increase in the weight. F)For any increase in cost of the candy mix the corresponding increase in the weight is 1/3 times that increase in the cost. H) If the weight of the candy mix varies from x=1 to x=7 pounds, then the cost increases by 18 dollars. I)y=3x+1.5

The function rv gives the rate of change of the volume of oil expelled per day by a rig that begins leaking at day t=0. The function r gives the approximate rate function with moments of duration 0.3 days for 0≤ t ≤1.5. We use the rates at the left endpoint of each moment to define the constant rate for that moment. Consequently the graph of an approximate rate function is a step function of piece-wise horizontal line segments over each moment of duration Δt=0.3 days. Check the statements that are TRUE and only those. A)r(0.4)=rv(0.4) B)r(t)=rv(0.3) when 0 ≤ t < 0.3 C)r(t)=rv(t) when 0 ≤ t ≤ 0.3 D)r(t)=rv(0.9) when 0.9 ≤ t < 1.2 E)r(1.4)=rv(1.2) F)r(0.4)=rv(0.3) G)r(t)= rv(0.6) when 0.6 ≤ t < 0.9 H)r(t)= rv(1.2) when 1.2 ≤ t < 1.5

D)r(t)=rv(0.9) when 0.9 ≤ t < 1.2 E)r(1.4)=rv(1.2) F)r(0.4)=rv(0.3) G)r(t)= rv(0.6) when 0.6 ≤ t < 0.9 H)r(t)= rv(1.2) when 1.2 ≤ t < 1.5

An online movie website charges a one-time membership fee, plus a rate of m = $1.49 per movie. If we use the symbols dy = m*dx to represent the given information, which is a possible meaning for dy? A)the total number of movies watched B)The total cost in dollars for joining the site and watching some number of movies C)a variation in the number of movies watched D)the variation in cost, in dollars, if you watch some more movies.

D)the variation in cost, in dollars, if you watch some more movies.

Given a variable y, which of the following calculates dy, a variation in y? A) yfinal/yinitial B) yinitial/yfinal C)yinitial+yfinal D) yinitial-yfinal E) yfinal-yinitial

E) yfinal-yinitial

A rocket-powered basketball was launched from the ground. The rocket's velocity was measured, in meter per second, for the first 10 seconds after the launch. The rocket's velocity function v is given by v(t)=t2, 0 ≤ t ≤ 10. The height of the rocket above the ground 2 seconds after the launch is 2.67 meters. Which expression calculates the approximate height of the rocket (total approximate accumulation in the rockets height) above the ground at t=4 seconds after the launch using Δt=0.8 in the approximation. A)22*0.8+2.82 *0.8+3.62*0.4 B)None of these are correct C)2.67+22*0.8+2.82 *0.8+3.62*0.8 D)2.67+02*0.8+0.82 *0.8+1.62*0.4 E)2.67+22*0.8+2.82 *0.8+3.62*0.4 F)2.67+22*2.8+2.82 *3.6+3.62*4

E)2.67+22*0.8+2.82 *0.8+3.62*0.4

A rocket-powered basketball was launched from the ground. The rocket's velocity was measured, in meter per second, for the first 10 seconds after the launch. The rocket's velocity function v is given by v(t)=t2 , 0 ≤ t ≤ 10. The height of the rocket above the ground 2 seconds after the launch is 2.67 meters. Which expression calculates the approximate net accumulation in rocket's height between t=2 seconds and t=4 seconds since the launch using Δt=0.8 in the approximation. A)None of these answers are correct B)0^2*0.8+0.82 *0.8+1.62*0.4 C)2^2*0.8+2.82 *0.8+3.62*0.8 D)2.67+22*0.8+2.82 *0.8+3.62*0.4 E)2^2*0.8+2.82 *0.8+3.62*0.4 F)2.67+22*2.8+2.82 *3.6+3.62*4

E)2^2*0.8+2.82 *0.8+3.62*0.4

Given a variable y, which of the following determines a final value of y? A)yinitial - dy B)yfinal - dy C)yfinal - yinitial D)dy E)yinitial+dy

E)yinitial+dy

T or F If for every unit change, dx, in x the corresponding change, dy, in y is 2, then y varies at a constant rate 2 with respect to x.

F

T or F If y varies at a constant rate 2 with respect to x, then y=2x is always true.

F

T or F If y=2x+1, then dy=2dx+1 for any variation, dx, in x and corresponding variation, dy, in y. True

F

T or F If for any variation, dx, in x the corresponding variation, dy, in y is 2 times that variation in x (that is dy=2dx) then y varies at a constant rate 2 with respect to x.

T

T or F If the value approximate rate of change function r at the moment x=a is zero, then approximate net accumulation function A function is neither increasing nor decreasing, the graph is a horizontal line segment in that moment.

T

T or F If the value of approximate rate of change function r at the moment x=a is negative number m, then approximate net accumulation function A function is a decreasing linear function with constant rate (slope) m in that moment.

T

T or F If the value of approximate rate of change function r at the moment x=a is positive number m, then approximate net accumulation function A function is an increasing linear function with constant rate (slope) m in that moment.

T

T or F If y varies at a constant rate 2 with respect to x, then for a unit change in x the corresponding variation in y is 2.

T

T or F If y varies at a constant rate 2 with respect to x, then for any variation, dx, in x the corresponding variation, dy, in y is 2 times that variation in x. Symbolically, dy=2dx.

T

T or F If y varies at a constant rate 2 with respect to x, then y=2x+c for some real number c.

T

T or F If y=2x+1, then dy=2dx for any variation, dx, in x and corresponding variation, dy, in y.

T

T or F If y=2x, then dy/dx=2 for any variation, dx, in x and corresponding variation, dy, in y.

T