Calculus Concepts and Applications

Vertical Motion A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. A) Find the position function giving the height s as a function of the time t. B) When does the ball hit the ground?

Building a distance function At time t = 0, a car travelling with velocity 96 ft/s begins to slow down with constant deceleration ? = 12 ??/?2. Find the velocity v(t) at time t and the distance traveled before the car comes to a halt. No physics equations necessary

Second Order Integral Find f if f x = 12?2+ 6? 4, ?? ? 0 = 4 And f (1) = 1

The Fundamental Theorem of Calculus

Branches of Calculus Differential Calculus - Core Concept: Tangent Line Problem Integral Calculus - Core Concept: Area Problem The Fundamental Theorem of Calculus Relates these two field of Calculus

Informal FTC Differentiation and (Definite) Integration are inverse operations in the same sense that division and multiplication are inverse operations. Slope of secant line is a quotient Area of the rectangle is a product Limit processes to get the slope of tangent line and Area under Curve preserve this invers reationship.

Formal Statement (Part 1) If a function f is continuous on the closed interval [ a , b ], then ? ? ? ?? = ? ? ?(?) ? Aka the process of calculating a definite integral

Proof! If we split the interval [a,b] into many sub- intervals (x0 , x1, x2, x3, x4, xn-1, xn,) We can subtract each pair and add all of the pairs together: ? ?? ?(?? 1) + ? ?? 1 ?(?? 2)+ + ? ?2 ? ?1+ ? ?1 ?(?0) If we combine like terms we get ? ?? ?(? 1) aka ? ? ?(?)

More Proof This expansion can also be represented by ? ? ? ?(?) = [? ?? ?(?? 1)] ?=1 Mean Value Theorem tells us that there exists some number ci in the ith subinterval such that ? ?? =? ?? ?(?? 1) ?? ?? 1

Last Steps ? ?? = ? ?? and ??= ?? ?? 1 We can rewrite to get ? ? ? ? ? ? = ? ?? ?? ?(?)?? ? ?=1

FTC: A Users Manual Provided you can find an antiderivative we can evaluate a definite integral as we have been doing to find a sum. When applying the FTC the following notation is convenient: a b ? f x dx=F x = ? ? ?(?) ? It is not necessary to include the constant of integration because it will cancel.

Mean Value Theorem for Integrals If f is continuous on the closed interval [a , b], then there exists a number c in the closed interval such that: ? ? ? ?? = ?(?)(? ?) ? What it means: There is some rectangle that has the same area as the curve which is somewhere in between the area of the inscribed rectangle and the Circumscribed Rectangle. This is an existence theorem. It does not tell us how to find this rectangle only that it exists.

Value from MVT for Integrals Average value of a Function: If f is integrable on the closed interval [a, b], then the average value of f on the interval is ? 1 ? ? ? ? ?? ?

Find the Average Value of a Function Ex. Find the average value of ? ? = 3?2 2? on the interval [1 , 4]

The Speed of Sound At different altitudes in the Earth s atmosphere, sound travels at different speeds. The speed of sound s(x) can be modeled by the piecewise function below where x is altitude in kilometers. What is the average speed of sound over the interval [0, 80]? 4? + 341 [0,11.5] 295 3 4? + 278.5 [22,32] 3 2? + 253.5 3 2? + 404.5 [50,80] [11.5,22] ? ? = [32,50]

Integral as a Function Definite integral can be a function of x ? ? ? ? ?? ??. ? ? ?? ? ?

Evaluate ? ??? ? ?? 0 At x = 0, ? 6,? 4,? 3,? 2,

The Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then for every x in the interval. ? ? ?? ? ? ?? = ?(?) ?

Evaluate ? ? ?? ?2+ 1?? 0

?3 ? ?? ? cos??? 2

Hw pg. 327 51, 53, 55, 57, 59, 73, 74, 75 87,89,91,93, 95, 97, 99