Graph of Flow Rate Versus Time

Graph of Flow Rate Versus Time
𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

The x variable (t) is to the first power. 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

The x variable (t) is to the first power. 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 The graph will be a straight line. Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time The line will have a positive slope.
𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 The line will have a positive slope. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 This will be the y-intercept of the line. Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time Here is a line with a positive slope.
𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time Here is a line with a positive slope. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time Here is the y-intercept.
𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time Here is the y-intercept. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time This is the point at which the drainage ceases. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time This is the point at which the drainage ceases. 𝐝𝐯 𝐝𝐭 =𝟎 and 𝐭= 𝐓 𝐅 − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time 𝟎=− 𝟐 𝐇 𝟎 𝐂 + 𝟐 𝐓 𝐅 𝐂 𝟐 𝐀 This is the point at which the drainage ceases. 𝐝𝐯 𝐝𝐭 =𝟎 and 𝐭= 𝐓 𝐅 − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time 𝟎=− 𝟐 𝐇 𝟎 𝐂 + 𝟐 𝐓 𝐅 𝐂 𝟐 𝐀 This is the point at which the drainage ceases. 𝐓 𝐅 =𝐂𝐀 𝐇 𝟎 𝐝𝐯 𝐝𝐭 =𝟎 and 𝐭= 𝐓 𝐅 − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

T F 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time Note that the drainage is negative ─ out of the Funnel. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

T F 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time Note that the equation would have the Funnel fill back up with water after drainage ends. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time We know this does not happen.
T F 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time Note that the equation would have the Funnel fill back up with water after drainage ends. − 2 H 0 C We know this does not happen. dv/dt Calculus – as Used by Scientists and Engineers by Graube

T F 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Time Therefore we are only using a fragment of the full function. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

T F Volume= Flow Rate Time Time − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time This axis is the flow rate.
T F Volume= Flow Rate Time Time This axis is the flow rate. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

T F Volume= Flow Rate Time Time This axis is the time. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

T F Volume= Flow Rate Time Time Therefore the area under the curve is the volume of water that has drained out of the Funnel. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

t T F Time This is the clock time. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time This is the time variable.
t T F Time This is the time variable. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

t T F Time Here is the time variable extension to the function. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

t T F Time This is the volume of water that has drained from the Funnel at time t. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

t T F Time This is the volume of water that has yet to drain from the Funnel at time t. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

t T F Time As time moves from left to right, the drain rate slows as the Funnel empties. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

Calculation of Exhaust Volume versus Time

This is the exhaust volume we are seeking. Calculus – as Used by Scientists and Engineers by Graube

12 32 T F Time − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

12 32 T F Time This is the exhaust volume we are seeking. − 2 H 0 C dv/dt Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

Here is the volume. 𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐯 𝐝𝐭 𝐝𝐭 = − 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐭 Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐯 𝐝𝐭 𝐝𝐭 = − 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐭 𝐝𝐯 𝐝𝐭 𝐝𝐭 =𝐝𝐯 𝐝𝐯= − 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐭 Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯 𝐝𝐭 =− 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐯 𝐝𝐭 𝐝𝐭 = − 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐭 𝐝𝐯 𝐝𝐭 𝐝𝐭 =𝐝𝐯 𝐝𝐯= − 𝟐 𝐇 𝟎 𝐂 + 𝟐𝐭 𝐂 𝟐 𝐀 𝐝𝐭 𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 We now have the volume isolated. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 However, it is the differential volume. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 How can we get to “real” macro volume? Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 We get “real” macro volume by integrating the differentials. Calculus – as Used by Scientists and Engineers by Graube

Rules of Integration − 𝐓𝐡𝐢𝐬 𝐢𝐬 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐬𝐢𝐠𝐧
The integral sign represents a stretched-out “S”. The “S” stands for “Sum”. Integration adds together the thin slices of a variable. Calculus – as Used by Scientists and Engineers by Graube Demonstration

Calculus – as Used by Scientists and Engineers by Graube
Rules of Integration Each integral must have a differential in its integrand upon which the integral does its job. Each integral works on only one variable at a time. An integral operates only on variables. 𝐌 𝐝𝐱 =𝐌 𝐝𝐱 𝐚 𝐝𝐱+𝐛 𝐝𝐱 = 𝐚 𝐝𝐱+ 𝐛 𝐝𝐱 Each integral must be given a start and a finish. These are called “limits of integration”. Calculus – as Used by Scientists and Engineers by Graube

Definition of start and finish for an integral.
Rules of Integration Definition of start and finish for an integral. Start Calculus – as Used by Scientists and Engineers by Graube

Rules of Integration Definition of start and finish for an integral. Finish Start Calculus – as Used by Scientists and Engineers by Graube

𝐒𝐭𝐚𝐫𝐭 𝐅𝐢𝐧𝐢𝐬𝐡 Rules of Integration
Definition of start and finish for an integral. 𝐒𝐭𝐚𝐫𝐭 𝐅𝐢𝐧𝐢𝐬𝐡 The variable travels from start to finish. Calculus – as Used by Scientists and Engineers by Graube

Rules of Integration Definition of start and finish for an integral. 𝐒𝐭𝐚𝐫𝐭 𝐅𝐢𝐧𝐢𝐬𝐡 The integral gives you the distance traveled by the variable from start to finish. Calculus – as Used by Scientists and Engineers by Graube

Refresher on Travel Distance
Travel is a vector. Calculus – as Used by Scientists and Engineers by Graube

Travel is a vector. Each trip has a start and a finish. Finish Start Calculus – as Used by Scientists and Engineers by Graube

Finish Start Travel 𝐃𝐢𝐬𝐭𝐚𝐧𝐜𝐞 ≡ 𝐏𝐨𝐬𝐢𝐭𝐢𝐨𝐧 − 𝐏𝐨𝐬𝐢𝐭𝐢𝐨𝐧 Finish Start Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 We get “real” macro volume by integrating the differentials. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐝𝐯 = − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐝𝐯 = − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 We need to define the limits of integration. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐝𝐯 = − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Each integral will have its own unique set of limits. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐝𝐯 = − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 The left side integral will have limits with respect to volume, since the integral works on the variable of volume. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐝𝐯 = − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 The right side integral will have limits with respect to time, since the integral works on the variable of time. Calculus – as Used by Scientists and Engineers by Graube

This is the exhaust volume we are seeking. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐛 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 At the start, the water in the Funnel will be at 𝐕 𝐛 and the clock will read 12 seconds. Calculus – as Used by Scientists and Engineers by Graube

𝐝𝐯=− 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 At the finish, the water in the Funnel will be at 𝐕 𝐚 and the clock will read 32 seconds. Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

Calculation of Exhaust Volume versus Time Take the left side first.
𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Take the left side first. Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐚 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 =𝐯 𝐕 𝐛 Using a scientific calculator: Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐚 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 =𝐯 𝐕 𝐛 This bar with the limits of integration indicates that the result of the integration needs to calculate the distance of the journey of the volume. Calculus – as Used by Scientists and Engineers by Graube

Finish Start Travel 𝐃𝐢𝐬𝐭𝐚𝐧𝐜𝐞 ≡ 𝐏𝐨𝐬𝐢𝐭𝐢𝐨𝐧 − 𝐏𝐨𝐬𝐢𝐭𝐢𝐨𝐧 Finish Start Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐚 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 =𝐯 𝐕 𝐛 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝐕 𝐚 − 𝐕 𝐛 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐚 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 =𝐯 𝐕 𝐛 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝐕 𝐚 − 𝐕 𝐛 =∆𝐕 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝐚 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 =𝐯 𝐕 𝐛 𝐕 𝐛 𝐕 𝐚 𝐝𝐯 = 𝐕 𝐚 − 𝐕 𝐛 =∆𝐕 ∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 We will now deal with the right side of the equation. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Take this term first. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. Using a scientific calculator: Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. =𝟑𝟐 𝐒𝐞𝐜−𝟏𝟐 𝐒𝐞𝐜 𝟏𝟐 𝐒𝐞𝐜. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. =𝟑𝟐 𝐒𝐞𝐜−𝟏𝟐 𝐒𝐞𝐜=𝟐𝟎 𝐒𝐞𝐜 𝟏𝟐 𝐒𝐞𝐜. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. =𝟑𝟐 𝐒𝐞𝐜−𝟏𝟐 𝐒𝐞𝐜=𝟐𝟎 𝐒𝐞𝐜 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟐𝟎 𝐒𝐞𝐜 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. =𝟑𝟐 𝐒𝐞𝐜−𝟏𝟐 𝐒𝐞𝐜=𝟐𝟎 𝐒𝐞𝐜 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟐𝟎 𝐒𝐞𝐜 =− 𝟐 𝟐𝟔 𝐜𝐦 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐𝟎 𝐒𝐞𝐜 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. =𝟑𝟐 𝐒𝐞𝐜−𝟏𝟐 𝐒𝐞𝐜=𝟐𝟎 𝐒𝐞𝐜 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟐𝟎 𝐒𝐞𝐜 =− 𝟐 𝟐𝟔 𝐜𝐦 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐𝟎 𝐒𝐞𝐜 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐝𝐭=𝐭 𝟑𝟐 𝐒𝐞𝐜. =𝟑𝟐 𝐒𝐞𝐜−𝟏𝟐 𝐒𝐞𝐜=𝟐𝟎 𝐒𝐞𝐜 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 = − 𝟐 𝐇 𝟎 𝐂 𝟐𝟎 𝐒𝐞𝐜 =− 𝟐 𝟐𝟔 𝐜𝐦 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐𝟎 𝐒𝐞𝐜 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Take this term next. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. Using a scientific calculator: Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 𝟏𝟐 𝐒𝐞𝐜. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 = 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟐𝟖.𝟓 𝐜𝐦 𝟐 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 = 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟐𝟖.𝟓 𝐜𝐦 𝟐 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 =𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 = 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟐𝟖.𝟓 𝐜𝐦 𝟐 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 =𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 +𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 = 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟐𝟖.𝟓 𝐜𝐦 𝟐 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 =𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 +𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 =−𝟑𝟐𝟗.𝟖 𝐜𝐦 𝟑 Calculus – as Used by Scientists and Engineers by Graube

∆𝐕= 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 + 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝐭 𝐝𝐭= 𝐭 𝟐 𝟐 𝟑𝟐 𝐒𝐞𝐜. = 𝟑𝟐 𝐒𝐞𝐜 𝟐 𝟐 − 𝟏𝟐 𝐒𝐞𝐜 𝟐 𝟐 =𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 𝟏𝟐 𝐒𝐞𝐜. 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 = 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐒𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟐𝟖.𝟓 𝐜𝐦 𝟐 𝟒𝟒𝟎 𝐒𝐞𝐜 𝟐 =𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 𝟏𝟐 𝐒𝐞𝐜. 𝟑𝟐 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =−𝟓𝟕𝟔.𝟖 𝐜𝐦 𝟑 +𝟐𝟒𝟕.𝟎 𝐜𝐦 𝟑 =−𝟑𝟐𝟗.𝟖 𝐜𝐦 𝟑 ∆𝐕=−𝟑𝟐𝟗.𝟖 𝐜𝐦 𝟑 Calculus – as Used by Scientists and Engineers by Graube

This is the exhaust volume we are seeking. ∆𝐕=−𝟑𝟐𝟗.𝟖 𝐜𝐦 𝟑 Demonstration

Introduction to Error Analysis
Calculus – as Used by Scientists and Engineers by Graube

Calculation of Remaining Volume versus Time
𝐝𝐯 = − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝟎 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 At the start, the Funnel is full and the clock reads zero. Calculus – as Used by Scientists and Engineers by Graube

Real-time variables inserted. 𝐕 𝟎 𝐯 𝟏 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝟎 𝐯 𝟏 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝟎 𝐯 𝟏 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐕 𝟎 𝐯 𝟏 𝐝𝐯 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝟎 𝐯 𝟏 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐯 𝟏 𝐕 𝟎 𝐯 𝟏 𝐝𝐯 =𝐯 𝐕 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝟎 𝐯 𝟏 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐯 𝟏 𝐕 𝟎 𝐯 𝟏 𝐝𝐯 =𝐯 = 𝐯 𝟏 − 𝐕 𝟎 𝐕 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐕 𝟎 𝐯 𝟏 𝐝𝐯 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐯 𝟏 𝐕 𝟎 𝐯 𝟏 𝐝𝐯 =𝐯 = 𝐯 𝟏 − 𝐕 𝟎 𝐕 𝟎 𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 =− 𝟐 𝐇 𝟎 𝐂 𝐭 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 =− 𝟐 𝐇 𝟎 𝐂 𝐭 =− 𝟐 𝐇 𝟎 𝐂 𝐭−𝟎 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 =− 𝟐 𝐇 𝟎 𝐂 𝐭 =− 𝟐 𝐇 𝟎 𝐂 𝐭−𝟎 =− 𝟐 𝐇 𝟎 𝐂 𝐭 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 =− 𝟐 𝐇 𝟎 𝐂 𝐭 =− 𝟐 𝐇 𝟎 𝐂 𝐭−𝟎 =− 𝟐 𝐇 𝟎 𝐂 𝐭 𝟎 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 − 𝟎 𝟐 𝟐 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 − 𝟎 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 − 𝟎 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 = 𝐭 𝟐 𝐂 𝟐 𝐀 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝐭 𝟎 𝐒𝐞𝐜. 𝐭 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 − 𝟎 𝟐 𝟐 = 𝟐 𝐂 𝟐 𝐀 𝐭 𝟐 𝟐 = 𝐭 𝟐 𝐂 𝟐 𝐀 𝟎 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 − 𝐕 𝟎 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 − 𝐕 𝟎 = 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 𝟎 𝐒𝐞𝐜. 𝐭 − 𝟐 𝐇 𝟎 𝐝𝐭 𝐂 + 𝟐𝐭 𝐝𝐭 𝐂 𝟐 𝐀 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 − 𝐕 𝟎 =− 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

Calculation of Height versus Time
𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐯 𝟏 = 𝐡 𝟏 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐯 𝟏 = 𝐡 𝟏 𝐀 𝐯 𝟏 𝐀 = 𝐡 𝟏 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐯 𝟏 = 𝐡 𝟏 𝐀 𝐯 𝟏 𝐀 = 𝐡 𝟏 𝐡 𝟏 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐯 𝟏 = 𝐡 𝟏 𝐀 𝐯 𝟏 𝐀 = 𝐡 𝟏 𝐡 𝟏 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐕 𝟎 = 𝐇 𝟎 𝐀 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐯 𝟏 = 𝐡 𝟏 𝐀 𝐯 𝟏 𝐀 = 𝐡 𝟏 𝐡 𝟏 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐕 𝟎 = 𝐇 𝟎 𝐀 𝐕 𝟎 𝐀 = 𝐇 𝟎 Calculus – as Used by Scientists and Engineers by Graube

𝐯 𝟏 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐯 𝟏 𝐀 = 𝐕 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝐀 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐯 𝟏 = 𝐡 𝟏 𝐀 𝐯 𝟏 𝐀 = 𝐡 𝟏 𝐡 𝟏 = 𝐕 𝟎 𝐀 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐕 𝟎 = 𝐇 𝟎 𝐀 𝐕 𝟎 𝐀 = 𝐇 𝟎 𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time Height has units of 𝐇 𝟎 . Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time has units of 𝐓 𝐅 . Time Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time This is the point at which drainage starts: 𝐡 𝟏 = 𝐇 𝟎 and 𝐭=𝟎. Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time This is the point at which drainage finishes: 𝐡 𝟏 =𝟎 and 𝐭= 𝐓 𝐅 . Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time This portion of the parabola occurs before the clock starts, so it is meaningless. Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height This portion of the parabola occurs after drainage finishes, so it is also meaningless. Time Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Height Time Reality dictates that the curvature of water height versus time takes on the shape of this fragment of the parabola. Calculus – as Used by Scientists and Engineers by Graube

We now have a tool with which to probe 𝐡 𝟏 over time. 𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Calculus – as Used by Scientists and Engineers by Graube

Demonstration of Height versus Time
𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 Calculus – as Used by Scientists and Engineers by Graube Demonstration

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐀= 𝟑.𝟏𝟓 𝐜𝐦 𝟔.𝟏𝟓 𝐜𝐦 Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐀= 𝟑.𝟏𝟓 𝐜𝐦 𝟔.𝟏𝟓 𝐜𝐦 𝐀=𝟏𝟗.𝟒 𝐜𝐦 𝟐 Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐀= 𝟑.𝟏𝟓 𝐜𝐦 𝟔.𝟏𝟓 𝐜𝐦 𝐀=𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝐇 𝟎 =𝟐𝟖.𝟏𝟓 𝐜𝐦 Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐀= 𝟑.𝟏𝟓 𝐜𝐦 𝟔.𝟏𝟓 𝐜𝐦 𝐀=𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝐇 𝟎 =𝟐𝟖.𝟏𝟓 𝐜𝐦 𝐡 𝟏 =𝟐𝟖.𝟏𝟓 𝐜𝐦− 𝟐 𝟐𝟖.𝟏𝟓 𝐜𝐦 𝐭 𝟎.𝟑𝟓𝟑𝟔 𝐬𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝐭 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐬𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝟐 Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 = 𝐇 𝟎 − 𝟐 𝐇 𝟎 𝐭 𝐂𝐀 + 𝐭 𝟐 𝐂 𝟐 𝐀 𝟐 𝐀= 𝟑.𝟏𝟓 𝐜𝐦 𝟔.𝟏𝟓 𝐜𝐦 𝐀=𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝐇 𝟎 =𝟐𝟖.𝟏𝟓 𝐜𝐦 𝐡 𝟏 =𝟐𝟖.𝟏𝟓 𝐜𝐦− 𝟐 𝟐𝟖.𝟏𝟓 𝐜𝐦 𝐭 𝟎.𝟑𝟓𝟑𝟔 𝐬𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝐭 𝟐 𝟎.𝟑𝟓𝟑𝟔 𝐬𝐞𝐜 𝐜𝐦 𝟓 𝟐 𝟐 𝟏𝟗.𝟒 𝐜𝐦 𝟐 𝟐 𝐡 𝟏 =𝟐𝟖.𝟏𝟓 𝐜𝐦− 𝐜𝐦 𝐒𝐞𝐜 𝐭 + 𝟎.𝟎𝟐𝟏𝟑 𝐜𝐦 𝐒𝐞𝐜 𝟐 𝐭 𝟐 Calculus – as Used by Scientists and Engineers by Graube

(cm) Time (Sec) 𝟑𝟔.𝟑 𝐒𝐞𝐜 𝐡 𝟏 =𝟐𝟖.𝟏𝟓 𝐜𝐦− 𝐜𝐦 𝐒𝐞𝐜 𝐭 + 𝟎.𝟎𝟐𝟏𝟑 𝐜𝐦 𝐒𝐞𝐜 𝟐 𝐭 𝟐 Calculus – as Used by Scientists and Engineers by Graube Demonstration

𝐡 𝟏 =𝟐𝟖.𝟏𝟓 𝐜𝐦− 𝐜𝐦 𝐒𝐞𝐜 𝐭 + 𝟎.𝟎𝟐𝟏𝟑 𝐜𝐦 𝐒𝐞𝐜 𝟐 𝐭 𝟐 +/- 10% swim lanes Real-time feedback Calculus – as Used by Scientists and Engineers by Graube

𝐡 𝟏 =𝟐𝟖.𝟏𝟓 𝐜𝐦− 𝐜𝐦 𝐒𝐞𝐜 𝐭 + 𝟎.𝟎𝟐𝟏𝟑 𝐜𝐦 𝐒𝐞𝐜 𝟐 𝐭 𝟐 +/- 10% swim lanes Real-time feedback This is an example of the unreasonable effectiveness of calculus. Calculus – as Used by Scientists and Engineers by Graube

Scientists and engineers use calculus as a tool.
Conclusion Scientists and engineers use calculus as a tool. Calculus – as Used by Scientists and Engineers by Graube

Graph of Flow Rate Versus Time

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Graph of Flow Rate Versus Time

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