EGR 1101 Unit 8 Lecture #1 The Derivative (Sections 8.1, 8.2 of Rattan/Klingbeil text) A Little History 1. 2. 3. 4. Seventeenth-century mathematicians faced at least four big problems that required new techniques: Slope of a curve Rates of change (such as velocity and acceleration) Maxima and minima of functions Area under a curve Slope We know that the slope of a line is defined as m y t (using t for the independent variable). Slope is a very useful concept for lines. Can we extend this idea to curves in general? 40 y1(t) = 3*t + 4 35 30 y 25 20 (4,16) (3,13) 15 (2,10) 10 (1,7) 5 0 0 0.5 1 1.5 2 2.5 t 3 3.5 4 4.5 5 Derivative We define the derivative of y with respect to t at a point P to be the limit of y/t for points closer and closer to P. In symbols: dy dt lim t 0 y t Alternate Notations There are other common notations for the derivative of y with respect to t. One notation uses a prime symbol (): y ( t ) dy dt lim t 0 y t Another notation uses a dot: y ( t ) dy dt lim t 0 y t Tables of Derivative Rules In most cases, rather than applying the definition to find a function’s derivative, we’ll consult tables of derivative rules. Two commonly used rules (c and n are constants): d (c ) 0 dt d dt ( t ) nt n n 1 Differentiation Differentiation is just the process of finding a function’s derivative. The following sentences are equivalent: “Find the derivative of y(t) = 3t2 + 12t + 7” “Differentiate y(t) = 3t2 + 12t + 7” Differential calculus is the branch of calculus that deals with derivatives. Second Derivatives When you take the derivative of a derivative, you get what’s called a second derivative. Notation: dy 2 d ( dt ) d y 2 dt dt Alternate notations: y ( t ) y ( t ) Forget Your Physics For today’s examples, assume that we haven’t studied equations of motion in a physics class. But we do know this much: Average velocity: v avg y t Average acceleration: a avg v t From Average to Instantaneous From the equations for average velocity and acceleration, we get instantaneous velocity and acceleration by taking the limit as t goes to 0. Instantaneous velocity: v (t ) dy dt Instantaneous acceleration: a (t ) dv dt Today’s Examples 1. 2. Velocity & acceleration of a dropped ball Velocity of a ball thrown upward Maxima and Minima Given a function y(t), the function’s local maxima and local minima occur at values of t where dy dt 0 Maxima and Minima (Continued) Given a function y(t), the function’s local maxima occur at values of t where dy 0 and dt 2 d y dt 2 0 Its local minima occur at values of t where dy dt 0 and 2 d y dt 2 0 EGR 1101 Unit 8 Lecture #2 Applications of Derivatives: Position, Velocity, and Acceleration (Section 8.3 of Rattan/Klingbeil text) Review Recall that if an object’s position is given by x(t), then its velocity is given by v ( t ) lim t 0 x t dx dt And its acceleration is given by a ( t ) lim t 0 v t dv dt 2 d x dt 2 Review: Two Derivative Rules Two commonly used rules (c and n are constants): d (c ) 0 dt d dt ( t ) nt n n 1 Three New Derivative Rules Three more commonly used rules ( and a are constants): d (sin( t )) cos( t ) dt d (cos( t )) sin( t ) dt d dt (e at ) ae at Today’s Examples 1. 2. 3. 4. 5. Velocity & acceleration from position Velocity & acceleration from position Velocity & acceleration from position (graphical) Position & velocity from acceleration (graphical) Velocity & acceleration from position Review from Previous Lecture Given a function x(t), the function’s local maxima occur at values of t where dx 0 and dt 2 d x dt 2 0 Its local minima occur at values of t where dx dt 0 and 2 d x dt 2 0 Graphical derivatives The derivative of a parabola is a slant line. The derivative of a slant line is a horizontal line (constant). The derivative of a horizontal line (constant) is zero.