Review for Test II: Rates, Derivatives, Optimization

1. Lecture 14 Review for Test II

2. Rate = Derivative Rate, rate of change, rate of growth, rate of decay, velocity, etc The amount A(t) to which principal P accumulates after t years at interest rate r per year (r a decimal) is: Simple Interest: A(t) = P(1+ rt) Compound Interest: Continuous Interest

3. Simple Interest • $1000 is borrowed for 5 years at 7% per year. • What amount is due after 5 years and, (b) at what rate • is the amount changing at that time? (b)

4. Compound Interest • $1000 is borrowed for 10 years at 9% per year compounded quarterly. • To what has the principal accumulated after 10 years and • at what rate is the amount due changing after 10 years

5. Continuous Interest • $1000 is borrowed at 9% compounded continuously for 10 years. • To what amount has the $1000 accumulated after 10 years • At what rate is the total accumulation changing after 10 years

6. Rates and Distance If a rock is thrown upward from a height of 10 feet with a velocity of 73 feet per second then in the absence of air resistance its height at time t will be What are is its velocity and acceleration when it hits the ground? Solve h(t) = 0 to get time when height is 0. t = -.133, t = 4.695 Velocity at time t is h ‘ ’ (4.6956) = -32 Acceleration at time t is h ‘’ (t) = -32

7. Rates and Distance II A particle moves on the x-axis so that its position at time t is At time t = 3 at what rate is its distance from the point (2,5) changing? Position at time t is Let d(t) = distance from (2,5) at time t

8. Geometry of the Derivative Must be able to infer properties of f(x) from properties of f ‘ (x) and f ‘ ’(x) Critical number Of f. Local max Of f occurs here Suppose this is the graph of the derivative of f. critical number of f Local min of f occurs here Critical number of f ‘ (f ‘’ = 0) – point of inflect occurs here Critical number of f ‘ (f ‘ ‘ = 0) a point of Inflection does NOToccur here. f dec f inc f dec f concave up f concave down

9. Geometry of Derivative II Suppose f ‘ (x) = (x-3)(x+1) what are: (a) the intervals on which f is increasing and decreasing, (b) the local maxima and minima of f, (c) the intervals on which f is concave up and down, (d) the points of inflection of the graph of f? f ‘’ (x) = 2-2x Concavity changesat x=1, flex point cp cp Change from inc to dec at x= -1 (local max) Change from dec to Inc at x= 3 (local min) pfp -1 3 1 Inc: (-inf,-1),(3, inf)dec: (-1.3)Concave down: (-inf,1) Concave up: (1, inf) f ‘ > 0 (inc) f ‘ < 0 (dec) f ‘ >0 (inc) f ‘’ < 0 (concave down) f ‘ ‘ > 0 (concave up)

10. Optimization If f is continuous on a closed interval [a,b] then f assumes its (absolute) maximum and (absolute) minimum values on [a,b] at critical numbers of f. • Strategy for finding absolute maxima and minima of f on [a,b]: • Find the critical numbers c1, c2, …, of f. Be sure to include the end points“a” and “b” among them. • Calculate the numbers f(c1), f(c2), …. The largest is the absolutemaximum and the smallest is the absolute minimum Note that without end points there may be no absolutemaximum or minimum. Local maxima and minima mayexist and will occur at critical numbers.

11. Maxima and Minima when there are no end points Even if there are no (or only one) end point absolute maxima and minima can be detected in case there is exactly one interior critical point. If there is a unique interior critical number then it must be the point at which the absolute maximum or absolute minimum occurs. In situations such as this the unique critical number is where the absolute max or min occurs. It will be a min if f ‘ <0 to the left and f ‘ >0 to the right, a max if ‘ > 0 to the left and f ‘ >0 to the right. Alternatively one can check concavity: concave up (f ‘’ <0) at the pointindicates local max, concave down (f ‘’ >0) indicates local min.

12. Example A particle travels in the plane so that at time t its x-coordinate is x(t) = 2t-3 and its y-coordinate is y(t) = 4-t. At which time is it closest to the origin? min cp Min dist = at t = 2 2 f ‘ > 0 (inc) f ‘ <0 (dec)