Chap. 3 MATH 1431-18319 Annalisa Quaini quaini@math.uh.edu Office : PGH 662 Lecture : MWF 11AM-12PM Office hours : W 8AM-10AM Daily quiz 6 is due on Friday at 11 AM. Exam 1 is coming - check the scheduler and registration on Courseware. http://www.math.uh.edu/ quaini A. Quaini, UH MATH 1431 1 / 11
Given h(x) = x 2 x 1. Find: the points of horizontal tangents. x such that h (x) < 0 and x such that h (x) > 0. A. Quaini, UH MATH 1431 2 / 11
The high order derivatives are obtained by taking the derivatives several times in a row... f (x), f (x), f (x), f (x), f (4) (x), f (5) (x),... f (x), d dx f (x), d 2 dx 2 f (x), d 3 dx 3 f (x), d 4 dx 4 f (x), d 5 f (x) =second derivative of f (x) f (x) =third derivative of f (x) dx 5 f (x),... A. Quaini, UH MATH 1431 3 / 11
3.4 Derivative As a Rate of Change A. Quaini, UH MATH 1431 4 / 11
Example : volume of a sphere V (r) = 4 3 πr 3. Rate of change of the volume of the sphere with respect to the radius is V (r) = 4πr 2. To get the instantaneous rate of change, we take the derivative with respect to time. A. Quaini, UH MATH 1431 5 / 11
Velocity and Acceleration Given the position as a function of time x(t), the instantaneous rate of change of the position is the velocity of the object: x (t) = v(t). The instantaneous rate of change of the velocity is the acceleration of the object: x (t) = v (t) = a(t) Other notations: v = dx dt, a = dv dt = d 2 x dt 2 A. Quaini, UH MATH 1431 6 / 11
Definition Speed is the absolute value of velocity. Speed at time t = v(t). A. Quaini, UH MATH 1431 7 / 11
We can look at the sign of the velocity to determine in which direction the object is going. Positive sign: the object is going towards the positive x; Negative sign: the object is going towards the negative x; We can look at the sign of velocity and acceleration to determine if an object is speeding up or slowing down. If the signs are the same, the object is speeding up. If the signs are different, the object is slowing down. A. Quaini, UH MATH 1431 8 / 11
We can look at the sign of the velocity to determine in which direction the object is going. Positive sign: the object is going towards the positive x; Negative sign: the object is going towards the negative x; We can look at the sign of velocity and acceleration to determine if an object is speeding up or slowing down. If the signs are the same, the object is speeding up. If the signs are different, the object is slowing down. A. Quaini, UH MATH 1431 8 / 11
Examples Chap. 3 Sect. 3.3 Sect. 3.4 1 An object moves along a coordinate line, its position at each time t 0 is given by x(t) = 5t t 3. Find the position, velocity, speed and acceleration at time t 0 = 3. 2 A particle is moving along a horizontal coordinate line according to the formula x(t) = t 3 6t 2. Find v(t) and a(t). When is the particle moving to the left? When is the acceleration negative? 3 If x(t) = 1 2 t4 5t 3 + 12t 2, find the velocity of the moving object when its acceleration is zero. A. Quaini, UH MATH 1431 9 / 11
Examples Chap. 3 Sect. 3.3 Sect. 3.4 1 An object moves along a coordinate line, its position at each time t 0 is given by x(t) = 5t t 3. Find the position, velocity, speed and acceleration at time t 0 = 3. 2 A particle is moving along a horizontal coordinate line according to the formula x(t) = t 3 6t 2. Find v(t) and a(t). When is the particle moving to the left? When is the acceleration negative? 3 If x(t) = 1 2 t4 5t 3 + 12t 2, find the velocity of the moving object when its acceleration is zero. A. Quaini, UH MATH 1431 9 / 11
Examples Chap. 3 Sect. 3.3 Sect. 3.4 1 An object moves along a coordinate line, its position at each time t 0 is given by x(t) = 5t t 3. Find the position, velocity, speed and acceleration at time t 0 = 3. 2 A particle is moving along a horizontal coordinate line according to the formula x(t) = t 3 6t 2. Find v(t) and a(t). When is the particle moving to the left? When is the acceleration negative? 3 If x(t) = 1 2 t4 5t 3 + 12t 2, find the velocity of the moving object when its acceleration is zero. A. Quaini, UH MATH 1431 9 / 11
Free Fall of an object Chap. 3 Sect. 3.3 Sect. 3.4 Neglecting the friction of the air, the position of an object in free fall is y(t) = y 0 + v 0 t 1 2 gt2, where y 0 is the initial position. v 0 is the initial velocity. g is the gravitational constant (32 feet per second per second, or 9.8 meters per second per second). A. Quaini, UH MATH 1431 10 / 11
Examples Chap. 3 Sect. 3.3 Sect. 3.4 An object is dropped from a height of 16 feet. If we neglect air friction, how long will it take for the object to hit the ground? Give the velocity of the object on impact. An object is launched from a height of 20 feet. If we neglect air friction, and it takes 10 seconds for the object to hit the ground. Give the initial upward velocity of the object and the downward velocity of the object on impact. A. Quaini, UH MATH 1431 11 / 11
Examples Chap. 3 Sect. 3.3 Sect. 3.4 An object is dropped from a height of 16 feet. If we neglect air friction, how long will it take for the object to hit the ground? Give the velocity of the object on impact. An object is launched from a height of 20 feet. If we neglect air friction, and it takes 10 seconds for the object to hit the ground. Give the initial upward velocity of the object and the downward velocity of the object on impact. A. Quaini, UH MATH 1431 11 / 11