Rectilinear Motion and Tangent Lines Alyssa Cobranchi AP Calculus

Rectilinear Motion Introduction  Rectilinear Motion is the movement of a particle on a straight line.  It is an application of the derivative of a function.  Some examples can include a race car moving along a straight track, an object thrown from the top of a building and falling straight down, or a ball thrown straight up and then falling straight down.

Position Versus Time Curve

s-axis As a biker moves forward, his position increases as time does so the graph is a positive slope. When the biker stands still to get a drink of water, the graph is a horizontal line because time is continuing to increase while the position does not. When the biker moves backward, the position decreases while time still increases.

Formulas for Rectilinear Motion  Given a function, we can use the derivative and the second derivative to find velocity and acceleration of a particle as it moves along a line.  The position of a particle is given as the function.

 First, you have to figure out what you are given and what you need to find.  If you are given the position formula and are asked to find the velocity, you find the derivative. For acceleration, you find the second derivative.  If you are given the acceleration formula, you have to find the integral to get velocity, and again to get position.

 To determine the direction of a particle, you need to use the velocity formula of the particle.  After you simplify the velocity formula, you have to find the zeros and make a sign line.  MAKE SURE YOU LABEL YOUR SIGN LINE AS V(T)  On the sign line, where it is positive, the particle is moving to the right, and where it is negative, it is moving to the left.  The points on the sign line are the points at which the particle is at rest.

Negative Positive Critical points where the equation is equal to zero; the particle is at rest An odd number of dotted lines indicate negative—the particle is moving to the left. A positive to negative switch in signs indicates a change in direction

+-+

Distance Traveled  In order to find the total distance a particle has traveled, you first need to find the velocity formula.  After you find the velocity, you have to make a sign line in order to see when the particle changes direction.  Next you have to check the points given in the problem and put them in the position formula. You also have to check any points within the interval that the particle changes direction.  Finally, calculate the total distance between the positions at the different positions.

Example

Solution + + -

Solution Continued

Using Integrals  If you are given the acceleration or velocity formula, almost always you will have to find the integral to solve the problem.  Sometimes you have to take the integral twice in order to get the position formula to find distance traveled.  When taking integrals, remember to add a constant, C, to the integral and solve for the constant using values given in the problem.  After C is found, you can write an equation for the velocity or position (if you repeat the process) and use these to solve for what the problem asks for.

Example

Solution Given: acceleration formula Needed: position formula

TRY ME!

Solution

The line is tangent to the graph at x f(x) Slope= f’(x)

 The tangent line of a curve models the direction of the particle at a given point.  The tangent line shares exactly one point with the curve.  The derivative of the curve is equal to the slope of the tangent line.

 The formula used when finding a tangent line is point-slope form. y value found when given x value is put into the function given x value Slope of the tangent line—derivative of the function

 To find a tangent line, you need to do several things.  First, you need the function.  You have to take the derivative of the function given.  Put the x-value given into the derivative found; the answer is m in the formula.

 Put the x-value given into the given function; the answer is y 1 in the formula. ◦ If you are given an ordered pair, this step is not necessary; just use the y- value given.  Plug in all values found into the formula of tangent lines.

Solution

TRY ME!

Solution

Normal Lines The process of finding normal lines is the same as finding tangent lines. The slope of tangent line is the same as the curve. Normal lines, however are perpendicular to the curve, therefore the slope of the normal line is the negative reciprocal of the slope of the curve.

Bibliography mo/pc/c02/read/main/c02x2_7.htm mo/pc/c02/read/main/c02x2_7.htm /APCalculus/ABQuizzes/QuizOptimization. pdf /APCalculus/ABQuizzes/QuizOptimization. pdf ©Alyssa Cobranchi 2011