INTUITION MATTERS! Applied Calculus in Engineering: Complete

Explore how radians and degrees measure circle angles, compute arc length with l = r theta, and convert between units using theta = pi/180 degrees, including 90 degrees example.

Explore how to obtain the inverses of sine, cosine, and tangent by rotating axes and using arcsin, arccos, and arctan, including quadrant context and domain ranges.

Convert 220 degrees to radians, compute arc length with r theta, and verify tangent ratios in a 45-degree triangle. Explore a real-life aerospace application of 1/x functions and multidimensional functions.

Analyze how to identify safe crossing intervals by tracing train edges and danger zones on a time graph, determining forbidden periods and free intervals to prevent collisions.

Apply a 1.1 second delay to the left car (purple at rest) to avoid the crash as it approaches the danger zone (blue) within the forbidden time interval.

Apply calculus intuition to prevent crashes by analyzing a two-term sinusoidal function with contrasting amplitude and frequency, ensuring the duration stays within the allowed time interval.

Tackle a tougher two-train simulation by accounting for blue and red trains and ensuring safe passage through both lines of danger zones to avoid crashes.

Present the solution to a tougher exercise, showing how to track four danger zones and two cars with blue and red trains, using a 2/2 delay to avoid crashes.

See how distance depends on time and wind in a two-variable function, forming a 3D surface with a two-dimensional domain.

Explore how distance depends on time, wind, and altitude, forming a four dimensional function beyond geometric graphing. Trust the math and visualize with domain output representations.

Explore how nonlinear functions behave near a point by comparing a function's value at x equals one with its limit as x approaches one from both sides, using intuitive examples.

Explore how €1,000 grows under different compounding schedules, including continuous compounding, illustrating present value, future value, inflation, and the power of compound interest.

Learn to model a wave using a sine function, where amplitude, frequency in hertz, and time determine radians via two pi f t, with A, B, and C shaping waves.

Explore derivatives using a tangent line to height-time curve at a moment. Apply delta h over delta t limits as delta t approaches zero to obtain instantaneous rate of change.

Explore how altitude changes over time on Earth, Mars, and the Moon by deriving downward velocity and downward acceleration from the position function using the limit method.

The lecture builds derivative intuition through three functions, highlighting continuity and smoothness as prerequisites for differentiability. It shows how discontinuities and a vertical tangent yield undefined or infinite derivatives.

Apply the chain rule to express theta as a function of y in a water pipe context, deriving d theta/dy = 1/(r cos theta) from sine theta = y/r.

Learn how constants have zero derivative and apply the power rule to derivatives of x^n, with proofs from the limit definition using delta x.

Demonstrate addition-subtraction and product rules with examples like y = x^2 ± x and y = x^2·x, and show how to replace the quotient rule using product and chain rules.

Develop intuition for applied calculus in engineering by applying derivative rules to a function with a denominator, using quotient or product forms and chain rule for dy/dx.

Explore derivatives of sine, cosine, and tangent with limits, showing that d/dx sin x equals cos x and d/dx cos x equals -sin x.

Learn to differentiate tangent theta by rewriting as sin theta over cos theta, applying product and chain rules, and simplifying to 1/cos^2 theta (sec^2 theta).

Apply derivatives to minimize a can's material under a fixed volume, deriving cylinder volume and surface area, locating critical points, and using the second derivative test to identify minima.

Minimize the can's surface area under a fixed volume by expressing height as a function of radius, differentiating to find the minimum, and using the second derivative test.

Use derivatives to maximize profit by formulating profit as revenue minus cost, finding critical points with the first derivative, and applying the second derivative test to identify a maximum.

Explore how vectors represent multi-dimensional quantities, from per-tank volumes to three-dimensional positions, and learn to project vectors onto x, y, and z planes.

Explore vector magnitude in three-dimensional space by computing the square root of the sum of squares of velocity components, and learn when magnitude reflects direction and strength.

Build a vector from its x and y components with i_hat and j_hat, using a grid to decompose and add components to obtain v = v_x i_hat + v_y j_hat.

Understand the cross product, which yields a vector perpendicular to the original vectors, unlike the dot product’s scalar, and apply the right hand rule with the exact three-dimensional formula.

Learn to model and control water levels in a three-tank storage building as a time-dependent vector, using functions of time f1(t), f2(t), f3(t) and parametric equations to drive pumps.

This lecture models three water tanks using piecewise functions, adding two cubic meters per second until each reaches 95 cubic meters, stopping at 22.5s, 27.5s, and 32.5s.

Three tanks start with 50, 40, 30 cubic meters; parabolic and cubic growth lead to time and x when volumes equal, using a difference function and quadratic formula.

This lecture builds three tank volume functions v1, v2, and v3 over time, solves x so v3 reaches 52+2√11 at t=1+√11, and visualizes the total volume with block diagrams.

Run a simulation to compare three water tanks, showing how different initial volumes yield the same final water level while tank three fills fastest.

Differentiate the tank volumes with respect to time to obtain volume flow, then multiply by density to get mass flow, elucidating instantaneous rates and units in kg per second.

Compute displacement vectors from position changes, derive velocity vectors by dividing by the time interval, and obtain acceleration vectors from velocity changes, using position vectors P0, P2, P3, P4.

Explain how displacement, velocity, and acceleration vectors relate as positions progress; show velocity follows displacement with varying magnitudes, and acceleration as the change of velocity, including tangent and perpendicular components.

Explore how acceleration arises from changes in velocity and momentum, including cases of changing mass like rockets, and distinguish tangential and normal components in circular motion.

Compute the derivative of a velocity vector by subtracting the old vector from the new and dividing by delta t, yielding acceleration for each component.

Explore Cartesian, cylindrical, and spherical coordinate systems and how grid elements represent space, while converting between x,y,z and r, theta, phi.

Convert from x and y to r and theta, expressing the same point in polar form. Use the Pythagorean theorem for r and tan theta = y/x, theta = arctan(y/x).

Explore a landing simulation that uses position vectors and displacement components to model airplane motion, clarifying the difference between velocity and speed in the X and Y dimensions.

Transform polar to Cartesian coordinates using layered global, polar, and x y prime coordinate systems to describe a 2D circular motion. Derive time-based position, velocity, and acceleration vectors.

Explore circular motion from synchronized cosine and sine functions, highlighting equal amplitudes to yield circles, and use vector addition and component-wise analysis to track position from origin.

Create a 3d spiral using cylindrical coordinates with a five-meter radius and z progression, and analyze how frequency and radius changes shape the motion, via Python simulations.

Vary the X and Y amplitudes with time to grow the spiral radius from linear to parabolic, creating a hurricane-like trajectory, and note when an amplitude exceeds the cosine amplitude.

Analyze a two-joint robot where both joints rotate with the same angular velocity, showing that total rotation relative to a global frame equals theta1 plus theta2, independent of link lengths.

Describe a point fixed to a rotating red frame from the blue frame by mapping red x prime, y prime to blue x, y coordinates using the shared position vector.

Express p3 in the global frame for a two-joint robot using a 2D rotation matrix with theta1 and theta2. Derive its velocity and acceleration in the xy1 frame.

Demonstrate a two-joint robot simulation with a changing first joint length and angles, showing how world and relative joint angles shape its trajectory as one hertz versus two hertz vary.

Derive the three-dimensional rotation matrices about the z, x, and y axes, show how to multiply them, and connect the two-dimensional cosine-sine basis to three-dimensional rotations.

Derive and apply three-dimensional rotation matrices for the x, y, and z axes, compose them to transform coordinates, and understand translational relations between reference frames.

Derive dy/dx for an implicit function by applying chain and product rules, using x^3 + y^3 - 9xy = 0 with x=2, y=4 to get 0.8.

Extend a simple function into higher dimensions by treating x and y as inputs and z as output, turning y = x^2 into a surface z = y - x^2.

Explore how adding inputs or outputs changes an implicit function: inputs form surfaces and violate the vertical plane test; adding a new output creates a higher-dimensional function with more information.

See partial derivatives in multivariable calculus reveal how distance changes with respect to time or altitude, holding the other variable constant.

Explore how X changes over time for a car moving on a spiral, using partial derivatives with respect to radius and angle, and compare to the direct time derivative.

Explore how the airplane's two-dimensional position magnitude changes with time and altitude using the chain rule for parametric x and y, driven by partial derivatives.

Model the airplane–mountain distance as 1000 minus X, where X depends on time and altitude, and apply the chain rule using partial derivatives for two inputs.

Identify the gradient as a five-element vector of first-level partial derivatives of y with respect to x1 through x5, and learn to apply the gradient operator.

Explore Taylor series as a method to approximate functions by constructing a shifted polynomial at point a, using derivatives and factorial terms, with sine as a key example.

Learn how Taylor series uses successive terms to approximate the sine function around a chosen point, from horizontal lines to linear, quadratic, and cubic polynomials.

Apply Taylor series to linearize a sine-based temperature model around 135 days, using the first two terms for a linear approximation, then add the second derivative for a quadratic approximation.

Estimate the area of a hangar wall shaped by a shifted parabola using delta x rectangles and a summation, then discuss reducing errors to approach the exact area.

Explore how differentiation loses constant information and how integration restores it only up to an arbitrary constant and initial condition, illustrating with x = t + c and accumulated area.

Explore the difference between indefinite and definite integrals by linking antiderivatives to accumulated area, using a cubic function and area under curves to turn functions into numbers.

Compute the area under y = 16 − (x − 4)^2 from x = 0 to 8 using substitution u = x − 4 to evaluate definite integral.

Apply definite integrals to find the wall area using substitution and a shifted parabolic function, yielding 85.3 square meters. The lecture contrasts positive and negative areas and previews improper integrals.

The lecture demonstrates a second approach to the area of a circle using a differential area and arc length, showing A = (1/2) R^2 ∫_0^{2π} dθ.

Apply the disk method by revolving the region under z = sqrt(y) about the y-axis, then integrate pi z^2 dy from 0 to 4 to get the volume.

Develop integration by parts by deriving its formula from the product rule and applying it to x cos x to obtain x sin x - cos x + C.

Compute the space arc length of a 3d spiral using parametric equations x=cos t, y=sin t, z=t and the speed integral from 0 to 2π to obtain 2π√2 meters.