# Calculus concepts and applications

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1 Calculus concepts and applications This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. The terms and conditions of this license allow for free copying, distribution, and/or modification of all licensed works by the general public. 1

2 Questions Question 1 Sketch the graphical interpretation of the derivative dx dot), and determine whether its value is positive or negative: +y at the point specified on the graph (the red -x +x -y file i

3 Question 2 Shade the area on this graph representing the following Riemann sum (assuming each horizontal and vertical division on the graph has an incremental value of 1): x y 9 n= 4 y x n Also, determine whether the numerical value of this Riemann sum is positive or negative. file i

4 Question 3 Shade the area on this graph representing the following integrals (assuming each horizontal and vertical division on the graph has an incremental value of 1): z x y w y dx w dx Also, determine whether the numerical value of this integral is positive or negative. file i

5 Question 4 Shade the area(s) on this graph representing the following integral (assuming each horizontal and vertical division on the graph has an incremental value of 1): w z y x 3 6 (w y) dx Also, determine whether the numerical value of this integral is positive or negative. file i

6 Question 5 Shade the area(s) on this graph representing the following integral (assuming each horizontal and vertical division on the graph has an incremental value of 1): w z y x (w y) dx Also, determine whether the numerical value of this integral is positive or negative. file i

7 Question 6 Write the integral expression represented by the shaded area on this graph (assuming each horizontal and vertical division on the graph has an incremental value of 1). The integral for this shaded area has a positive value: z x y w file i

8 Question 7 Write the integral expression represented by the shaded area on this graph (assuming each horizontal and vertical division on the graph has an incremental value of 1). The integral for this shaded area has a negative value: z x y w file i

9 Question 8 Write the integral expression represented by the shaded area on this graph (assuming each horizontal and vertical division on the graph has an incremental value of 1). The integral for this shaded area has a positive value: z x y w file i

10 Question 9 A small-scale biodiesel manufacturing plant records the production of biodiesel fuel by measuring the liquid level in a storage vessel. There is no flow transmitter monitoring flow rate into the vessel, but we can infer flow rate by monitoring vessel level over time. Level transmitter LT PV Computer LIU Ultrasonic Storage vessel Biodiesel The liquid level measurement signal coming from the ultrasonic level transmitter (LT) is our process variable (PV), and it is sent to a computer to be indicated and processed (LIU). The particular processing done in the computer is calculation of average flow between sample intervals. Suppose that the computer samples the transmitter s signal once every minute and records these measurements in a data file. Here is an example of that file s contents after one tank-filling batch, shown in a table format: Time Volume Time Volume Time Volume (minutes) (gallons) (minutes) (gallons) (minutes) (gallons)

11 Plotted over time on a graph, this data takes on the following form: Volume (gallons) Time (minutes) Just looking at this graph, what can you determine about the flow rate of biodiesel into this vessel? What would account for the flat spot in the middle of the graph? What do the minute variations in slope in the other areas of the graph represent, in terms of flow into the vessel? 11

12 Now, use the data in the table to calculate and plot the flow rate of biodiesel in gallons per minute into this vessel: Volume (gallons) Flow (gal/min) Time (minutes) How do these two plots relate to each other? That is, how does the shape of one refer to the shape of the other, geometrically? Suggestions for Socratic discussion Suppose a technician decided to calculate the flow rate at t = 4 minutes by taking the volume at that time (17.05 gallons) and dividing by the time (4 minutes). Explain why this would yield an incorrect value for flow, then explain the correct way to calculate flow rate at that time from the given data. file i

13 Question 10 Direct-chill aluminum casting is a process by which molten aluminum is cast into long billets. The aluminum is poured into a series of water-cooled moulds, solidifying as billets slowly lowered by the downward motion of a hydraulic ram. The motion of this ram is controlled by throttling oil flow out of the bottom, as the ram descends by gravity. Given a known cylinder diameter, the vertical speed of the ram will be a fixed proportion to the volumetric oil flow rate exiting it: Molten aluminum poured from furnace Direct-chill aluminum casting process Water-cooled mould Aluminum billet Aluminum billet Aluminum billet Aluminum billet Platen Slow motion Pit Hydraulic ram FIC M FT Oil reservoir However, if the piston seals within the ram leak, the ram s descent will not be perfectly proportional to oil flow rate. Consider a case where the normal rate of descent is 11 inches per minute, and the normal drop time is 25 minutes to cast billets 275 inches in length. With an empty platen, the descent from leakage is measured by a technician to be 0.3 inches per minute (with the hydraulic control valve shut). With a full platen (aluminum billets of full length), the descent from piston leakage is 1.7 inches per minute (with the hydraulic control valve shut). Calculate the actual length of the billets given an automatically-controlled flow equivalent to 11 inches per minute, and a drop time of 25 minutes. Assume the leakage rate grows linearly over time, from +0.3 inches per minute to +1.7 inches per minute, additional to the controlled flow of 11 inches per minute. 13

14 Suggestions for Socratic discussion A useful problem-solving technique for calculus-based problems is to sketch a graph of the variables being considered. The derivative of a function is the slope of the graph, while the integral of a function is the area bound by the graph. Apply this problem-solving technique to the problem at hand in this question. file i

15 Question 11 Complete a table showing x and y values for the following equation, then plot the equation on the coordinate grid provided: y = 3x + 2 x y y x Calculate the slope (at any point) for this equation. Explain both how you may determine the slope by looking at x and y values in the table, and also by examining the written equation. file i

16 Question 12 Unlike linear functions, the graphs of nonlinear functions do not have constant slopes. Take for instance this graph of a parabola of the form y = x 2 : y x It is difficult (but not impossible) to estimate the slope of a nonlinear function such as this at any given point, just by looking at the graph. If I were to ask you to estimate the slope of this equation s graph at x = 2, how would you do it? file i

17 Question 13 Two straight lines appear on this graph, along with a parabola, defined by the equation y = x 2. Dots mark where each of the straight lines intersects the parabola: y = x 2 One of these straight lines is a tangent line, while the other is a secant line. Identify which is which, and then give general definitions for each line type. file i

18 Question 14 Shown here is a view of a secant line and a tangent line (both thin) to a curve (thick): y curve x The slope of one of these lines is represented by the ratio of two increments, y and x: y x The slope of the other line is represented by the ratio of two differentials, and dx, better known as a derivative: dx Determine which slope ( y x dx ) belongs to which type of line (secant or tangent), and label all increments and differentials on the graph. file i01508 or 18

19 Question 15 Estimate the slope of a parabola (y = x 2 ) at x = 2 by calculating the slope of successive secant lines: First approximation (points of intersection x = 2 and x = 3) Slope = y x = y y x x 19

20 Second approximation (points of intersection x = 2 and x = 2.5) y Slope = y x = y x x 20

21 Third approximation (points of intersection x = 2 and x = 2.1) Slope = y x = y y x x 21

22 Fourth approximation (points of intersection x = 2 and x = 2.01) Slope = y x = y y x x Do you see a pattern here? Does the slope of the secant line tend to approach a certain value as we make x smaller and smaller? How does this secant line slope compare with the slope of the tangent line? file i

23 Question 16 Estimate the slopes of the tangent lines touching a parabola (y = x 2 ) at x = 3, x = 4, and x = 5 by using secant line approximations: If y = x 2 then, dx = x=3 dx = x=4 dx = x=5 Do you see a mathematical relationship between the value of x and the slope of the line tangent to that point on the parabola? Can you express that relationship as a function of x? file i01517 dx =??? 23

24 Question 17 Estimate the slopes of the tangent lines touching a nonlinear function (y = x 3 ) at x = 1, x = 2, x = 3, x = 4, and x = 5 by using secant line approximations: If y = x 3 then, dx = x=1 dx = x=2 dx = x=3 dx = x=4 dx = x=5 Do you see a mathematical relationship between the value of x and the slope of the line tangent to that point on the cubic function? Can you express that relationship as a function of x? file i01518 dx =??? 24

25 Question 18 The Power Rule in differential calculus tells us how to find a formula for the derivative of a function consisting of an integer power of the independent variable: If y = x n then dx = nxn 1 Using the Power Rule, differentiate the following functions: y = x dx = y = x 2 dx = y = x 3 dx = y = x 4 dx = y = x 5 dx = y = x 1 dx = y = x dx = file i01519 y = 11 dx = 25

26 Question 19 The Constant Multiple Rule in differential calculus tells us that the derivative of a function multiplied by a constant will be that constant multiplied by the derivative of the function: [ ] d If y = af(x) then dx = a dx f(x) The Sum Rule in differential calculus tells us that the derivative of a sum will be the sum of the derivatives: If y = f(x) + g(x) then dx = d dx f(x) + d dx g(x) Both of these rules (plus the Power Rule ) are useful when we must differentiate a polynomial function such as this: y = 3x 2 + 5x dx = 6x + 5 Apply these rules of differentiation to the following functions: y = 4x x 2 + 2x dx = y = x 5 + 4x 3 + 6x 2 dx = y = x 2 + 3x + 10 dx = y = 3x 2 8x 1 dx = file i01520 y = 5x x 2 6x dx = 26

27 Question 20 In physics, a very common formula relating position (x), velocity (v) and constant acceleration (a) is as follows: Where, x = Final position x o = Original position v o = Original velocity a = Acceleration t = Time x = x o + v o t at2 Differentiate this equation with respect to time, showing your work at each step. Also, explain the meaning of the derived equation (what does dx dt represent in physics?). file i01522 Question 21 Capacitors store energy in the form of an electric field. We may calculate the energy stored in a capacitance by integrating the product of capacitor voltage and capacitor current (P = IV ) over time, since we know that power is the rate at which work (W) is done, and the amount of work done to a capacitor taking it from zero voltage to some non-zero amount of voltage constitutes energy stored (U): P = dw dt dw = P dt U = W = P dt Find a way to substitute capacitance (C) and voltage (V ) into the integrand so you may integrate to find an equation describing the amount of energy stored in a capacitor for any given capacitance and voltage values. file i

28 Question 22 Inductors store energy in the form of a magnetic field. We may calculate the energy stored in an inductance by integrating the product of inductor voltage and inductor current (P = IV ) over time, since we know that power is the rate at which work (W) is done, and the amount of work done to an inductor taking it from zero current to some non-zero amount of current constitutes energy stored (U): P = dw dt dw = P dt U = W = P dt Find a way to substitute inductance (L) and current (I) into the integrand so you may integrate to find an equation describing the amount of energy stored in an inductor for any given inductance and current values. file i02465 Question 23 Calculate the output voltage of this opamp circuit assuming a constant input voltage of +1.5 volts applied for 0.5 seconds. Assume the capacitor begins in a state of zero charge (V 0 = 0 volts): +V kω 0.33 µf V in (volts) 1 + V out =??? Time (s) Also, calculate the time constant of this integrator circuit (i.e. the factor relating output voltage rate-of-change to input voltage). file i

29 Question 24 Calculate the output voltage of this opamp circuit given the input voltage profile shown in the graph. Assume the capacitor begins with a charge (V 0 ) of +8 volts: +V kω 2.2 µf V in (volts) 1 + V out =??? Time (s) Also, calculate the time constant of this integrator circuit (i.e. the factor relating output voltage rate-of-change to input voltage). file i

30 Question 25 Calculate the rate of liquid flow coming into process vessel V-5 at 1:30 PM, and also at 3:45 PM, based on the information shown here: FT 105 LT GPM 40 FT-240 V-5 30 FT 240 Gallons LT :00 1:00 2:00 3:00 4:00 5:00 Q 1:30 PM = Q 3:45 PM = GPM GPM file i

31 Question 26 Calculate the volume of liquid discharged from this vessel between 1:00 PM and 4:00 PM based on the information shown here: FT 105 LT GPM 40 FT-240 V-5 30 FT 240 Gallons LT :00 1:00 2:00 3:00 4:00 5:00 V discharged between 1:00 PM and 4:00 PM = gallons file i

32 Question 27 Calculate the amount of liquid lost from the vessel between 4:30 PM and 5:30 PM based on the information shown here: FT 105 LT GPM 40 FT-240 V-5 30 FT 240 Gallons LT :00 1:00 2:00 3:00 4:00 5:00 V lost between 4:30 PM and 5:30 PM = gallons file i

33 Question 28 The level transmitter (LT-31) on vessel V-5 failed at 1:00 PM today, its signal going all the way to zero even though there was still liquid inside the vessel. You are asked to calculate the amount of liquid in the vessel at 3:00 PM based on the flow trends shown (in units of gallons per hour): FT 105 LT FT 240 V-5 GPH Gallons FT-240 LT-31 FT-105 FT :00 1:00 2:00 3:00 4:00 5:00 FT-105 3:00 PM = gallons file i

34 Answer 1 Answers +y dx -x +x dx -y has a negative value at the red dot because the differential is negative while the differential dx is positive. 34

35 Answer 2 This Riemann sum has a positive value, because each rectangular area has a positive value (each one having a positive y value multiplied by positive x value): x y 35

36 Answer 3 Both integrals have negative values: z x y w The first integral (shaded red) has positive dx increments but negative y values. The second integral (shaded violet) has negative dx increments and positive w values. 36

37 Answer 4 This integral has a negative value: w z y x From x = 6 to approximately x = 3.5, the integral accumulates a positive value because it has positive dx increments and w is larger than y. From approximately x = 3.5 to x = 3 the integral accumulates a negative value because y is larger than w (a negative integrand) while dx increments still remain positive. Overall, the negative area is greater than the positive area, and so the integral from x = 6 to x = 3 has a net negative value. 37

38 Answer 5 This integral has a positive value: w z y x Answer 6 There are two different ways to write this integral: (w z) dx (z w) dx Answer 7 There are two different ways to write this integral: (8 y) dx (y 8) dx 38

39 Answer 8 There are two different ways to write this integral: (w + 7) dx (w + 7) dx Answer 9 Volume (gallons) Flow (gal/min) Time (minutes) 39

40 Answer 10 This is an integration problem, calculating distance (x) given velocity (v) and time (t). With a constant velocity, distance is the product of velocity and time: x = vt However, in this case we do not have a constant velocity. The velocity of the billets increases over time, due to the increasing platen weight and the piston leak. Distance traveled by the platen is best calculated using the following integral: x = v dt With no piston leakage, the integral may be expressed graphically as the area enclosed by a line 11 inches per minute high, and 25 minutes wide: v (in/min) 10 5 Area = (11 in/min)(25 min) = 275 inches t (min) 40

41 With leakage (starting at 0.3 inches per minute and progressing to 1.7 inches per minute), the integral area takes on a trapezoidal shape: v (in/min) t (min) The total area of this shape is 300 inches. Broken up into three distinct geometric shapes (two rectangles and one triangle): 25 min x = (11 in/min) dt = 275 inches 0 min 25 min x = (0.3 in/min) dt = 7.5 inches 0 min 25 min x = (0 to 1.4 in/min) dt = 17.5 inches 0 min Total distance traveled = 275 inches inches inches = 300 inches. 41

42 Answer 11 Slope = 3 y x 42

43 Answer 12 One way is to draw a line that just grazes the curve of the parabola at the coordinate (2,4) and estimate its slope: y (2,4) x Answer 13 A secant line intersects a curve at two points, approximating the slope of that curve at some point in between the two intersections. A tangent line, on the other hand, intersects a curve at a single point without crossing that curve, the slope of that line equaling the slope of that curve at the exact point of intersection. 43

44 Answer 14 y y x = dx curve x y = Slope of secant line x = Slope of tangent line dx Due to the classic definition of the derivative as being the limit of the secant line slope as x approaches zero, many people are initially led to believe that dx and must necessarily be very small. Not so! Differentials such as dx and may assume any non-zero real number value, it s just that they indicate the slope of the tangent line, not the slope of the curve between intervals. For practical applications of largevalued differentials, consult an introductory calculus book on the subject of linear approximations using differentials. 44

45 Answer 15 First approximation (points of intersection x = 2 and x = 3) Second approximation (points of intersection x = 2 and x = 2.5) Third approximation (points of intersection x = 2 and x = 2.1) Fourth approximation (points of intersection x = 2 and x = 2.01) Slope = y x = 5 Slope = y x = 4.5 Slope = y x = 4.1 Slope = y x = 4.01 Fine approximation (points of intersection x = 2 and x = ) Slope = y x = The slope of the secant line approaches 4 (which is the exact slope of the tangent line at x = 2) as we make x smaller and smaller. Answer 16 dx = 6 x=3 dx = 8 x=4 dx = 10 x=5 dx = 2x 45

46 Answer 17 dx = 3 x=1 dx = 12 x=2 dx = 27 x=3 dx = 48 x=4 dx = 75 x=5 dx = 3x2 46

47 Answer 18 y = x dx = 1 y = x 2 dx = 2x y = x 3 dx = 3x2 y = x 4 dx = 4x3 y = x 5 dx = 5x4 y = x 1 dx = x 2 y = x dx = 1 2 x y = 11 dx = 0 47

48 Answer 19 y = 4x x 2 + 2x dx = 12x2 + 34x + 2 y = x 5 + 4x 3 + 6x 2 dx = 5x4 + 12x x y = x 2 + 3x + 10 dx = 2x + 3 y = 3x 2 8x 1 dx = 6x 8 y = 5x x 2 6x dx = 15x2 + 20x 6 Answer 20 Showing my work, step-by-step: x = x o + v o t at2 d dt x = d dt x o + d dt v ot + d 1 dt 2 at2 d dt x = 0 + v o + at Where, dx dt represents velocity. dx dt = v o + at Answer 21 U = 1 2 CV 2 Answer 22 U = 1 2 LI2 48

49 Answer 23 V out = 1 tf V in dt + V 0 RC t 0 ( ) ( ) V out = ( Ω)( dt + 0 V F) 0 ( ) 1 V out = (0.75 V s) + 0 V s V out = V Calculating the time constant for this integrator circuit: τ i = RC = ( Ω)( F) = seconds 49

50 Answer 24 V out = 1 tf V in dt + V 0 RC t 0 ( 1 V out = ( Ω)( F) )( ) V in dt + 8 V At this point we need to evaluate the integral in order to proceed much further. Since V in is not a constant, and we have no means to symbolically integrate the input voltage function, we must find the integral value graphically. Recalling that the graphical meaning of integration is the geometric area encompassed by the function, all we need to do is calculate the area of the trapezoid: 2 Total area under the curve = V in (volts) V s = 3.15 volt-seconds V s 0.45 V s Time (s) Note the units of measurement used to express the integral: volt-seconds, because the vertical dimension is expressed in units of volts and the horizontal dimension is expressed in units of seconds and integration involves multiplication of units. ( ) 1 V out = (3.15 V s) + 8 V 0.22 s V out = V + 8 V V out = V Calculating the time constant for this integrator circuit: τ i = RC = ( Ω)( F) = 0.22 seconds Answer 25 At 1:30 PM, the level is increasing at a rate ( dv dt ) of 5 gallons per hour, which is equivalent to GPM. At that time the outgoing flow rate (FT-240) registers 35 GPM. Therefore, the incoming flow rate must be GPM greater than FT-240, which is GPM. Q 3:45 PM must be equal to Q out because the level (LT-31) is holding stea. Therefore, Q in = 27.5 GPM at 3:45 PM. What this means is that the incoming flow decreased at the same time as the outgoing flow decreased (both at 2:00 PM). Between Noon and 2:00 PM Q in was GPM and Q out was 35 GPM, but then both flow rates stepped down to 27.5 GPM at 2:00 PM. 50

51 Answer 26 Here we are asked to calculate a total volume given flow rate (in gallons per minute) and time. This involves multiplication (so that minutes of time will cancel out the minutes in GPM to yield an answer in gallons), which means the appropriate calculus function is integration. Specifically, we need to integrate the flow rate of FT-240 over the time interval of 1:00 PM to 4:00 PM: 4:00 PM V discharged = 1:00 PM Q FT 240 dt This integral represents the area beneath the FT-240 flow function between 1:00 PM and 4:00 PM on the trend graph, represented by the two shaded rectangles below: FT 105 LT GPM 40 FT-240 V-5 30 FT 240 Gallons gal 3300 gal LT :00 1:00 2:00 3:00 4:00 5:00 The first rectangle is 35 GPM high and 60 minutes wide, yielding an area of 2100 gallons: ( )( ) 35 gal 60 min = 2100 gal min 1 The second rectangle is 27.5 GPM high and 120 minutes wide, yielding an area of 3300 gallons: ( )( ) 27.5 gal 120 min = 3300 gal min 1 Together, the total area of these two rectangles is 5400 gallons, which is the value of our integral, and therefore the total quantity of liquid discharged from the vessel between 1:00 PM and 4:00 PM. V discharged between 1:00 PM and 4:00 PM = 5400 gallons 51

52 Answer 27 This question is designed to probe your critical thinking, because there is absolutely no calculus involved in the answer! Level transmitter LT-31 alrea measures the amount of liquid stored in the vessel, so calculating volume lost between any two points in time is simply a matter of subtracting those LT-31 values at those times: FT 105 LT GPM 40 FT-240 V-5 30 FT 240 Gallons LT gallons 0 12:00 1:00 2:00 3:00 4:00 5:00 Since the vessel holds 17.5 gallons of liquid at 4:30 PM and holds 10 gallons of liquid at 5:30 PM, the amount of liquid lost from the vessel between those times is 7.5 gallons: V lost between 4:30 PM and 5:30 PM = 7.5 gallons 52

53 Answer 28 We are asked to find the number of gallons inside the vessel given flow rates in gallons per hour and time in hours. Therefore, the mathematical operation we must employ is multiplication (so that hours cancels out to leave gallons ) and that means integration. The vessel s accumulated volume rises or falls according to the difference between the incoming and outgoing flow rates over time. Mathematically we may express this by the following integral: V = tf t 0 (Q in Q out ) dt + V 0 The level transmitter stopped working at 1:00 PM, so the last known volume of liquid inside the tank was measured then: 22.5 gallons. This must then be our initial volume (V 0 ), with 1:00 PM being the lower limit of our integration interval. The upper limit of our integration interval must be 3:00 PM which is the time when we re interested in the tank s liquid volume. Therefore: V 3:00 = 3:00 1:00 (Q in Q out ) dt gal Evaluating this integral graphically, we end up with the following shaded area between the two flowmeter trendlines: FT 105 LT FT 240 V-5 GPH Gallons FT-240 LT-31 FT gal FT :00 1:00 2:00 3:00 4:00 5:00 FT-105 The area of this trapezoid (calculated as the area of a 3x3 square and a 3x6 triangle) is (7.5 gal/hr)(0.5 hr) + (0.5)(7.5 gal/hr)(1 hr) = 7.5 gallons. Thus: V 3:00 = 3:00 1:00 (Q in Q out ) dt gal V 3:00 = 7.5 gal gal = 30 gal Vessel V-5 therefore contains 30 gallons of liquid at 3:00 PM. 53

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