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Section 2.2 – Basic Differentiation Rules and Rates of Change

Section 2.2 – Basic Differentiation Rules and Rates of Change. The Constant Rule. A constant function has derivative , or: Note: The constant function is a horizontal line with a constant slope of 0. Examples.

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Section 2.2 – Basic Differentiation Rules and Rates of Change

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  1. Section 2.2 – Basic Differentiation Rules and Rates of Change

  2. The Constant Rule A constant function has derivative , or: Note: The constant function is a horizontal line with a constant slope of 0.

  3. Examples The function is a horizontal line at y = 13. Thus the slope is always 0. The function is a horizontal line because e, Pi, and i all represent numeric values. Thus the slope is always 0. Differentiate both of the following functions.

  4. The Power Rule For any real number n, the power function has the derivative , or: Ex:

  5. Examples 1. Bring down the exponent 2. Leave the base alone 3. Subtract one from the original exponent. The procedure does not change with variables. Differentiate all of the following functions. Make sure the function is written as a power function to use the rule.

  6. The Constant Multiple Rule The derivative of a constant times a function, is the constant times the derivative of the function. In other words, if c is a constant and fis a differentiable function, then Objective: Isolate a power function in order to take the derivative. For now, the cf(x) will look like .

  7. Examples Take the derivative “Pull out” the coefficient Make sure the function is written as a power function to use the rule. Differentiate all of the following functions. Make sure the function is written as a power function to use the rule.

  8. The Sum/Difference Rule The derivative of a sum or a difference of functions is the sum or difference of the derivatives. In other words, if f and gare both differentiable, then OR Objective: Isolate an expressions in order to take the derivative with the Power and Constant Multiple Rules.

  9. Example 1 If find if and . Evaluate the derivative of k at x = 5 Find the derivative of k

  10. Example 2 Sum and Difference Rules Constant Multiple Rule Power Rule Evaluate: Simplify

  11. Example 3 First rewrite the absolute value function as a piecewise function Find the Left Hand Derivative Find the Right Hand Derivative Since the one-sided limits are not equal, the derivative does not exist at the vertex Find if .

  12. Example 4 Find the constants a, b, c, and d such that the graph of contains the point (3,10) and has a horizontal tangent line at (0,1). 1.f(x) contains the points (3,10) and (0,1) What do we know: 2. The derivative of f(x) at x=0 is 0 Use the points to help find a,b,c Find the Derivative We need another equation to find a and b We know the derivative of f(x) at x=0 and x=1 is 0

  13. Example 4 (Continued) Find the constants a, b, c, and d such that the graph of contains the point (3,10) and has a horizontal tangent line at (0,1). 1.f(x) contains the points (3,10) and (0,1) AND What do we know: 2. The derivative of f(x) at x=0 is 0 Use the Derivative Find a a = 1, b = 0, and c = 1

  14. Derivative of Cosine Using the definition of derivative, differentiate f(x) = cos(x). Find the derivative of sine if you have time

  15. Derivative of Sine Using the definition of derivative, differentiate f(x) = sin(x).

  16. Derivatives of Sine and Cosine

  17. Example 1 Rewrite the ½ to pull it out easier Rewrite the radical to use the power rule Sum and Difference Rules Constant Multiple Rule Power Rule AND Derivative of Cosine/Sine Differentiate the function: Simplify

  18. Example 2 Horizontal Lines have a slope of Zero. First find the derivative. Find the x values where the derivative (slope) is zero Find the corresponding y values Find the point(s) on the curve where the tangent line is horizontal.

  19. Calculus Synonyms The following expressions are all the same: • Instantaneous Rate of Change • Slope of a Tangent Line • Derivative DO NOT CONFUSE AVERAGE RATE OF CHANGE WITH INSTANTANEOUS RATE OF CHANGE.

  20. Position Function TIME: s(t) 2 0 3 1 2 5 6 4 7 2 Origin t 1 2 3 4 5 6 Description of Movement: -2 -2 -4 No Movement Downward Upward -4 The function s that gives the position (relative to the origin) of an object as a function of time t. Our functions will describe the motion of an object moving in a horizontal or vertical line.

  21. Displacement Displacement is how far and in what direction something is from where it started after it has traveled. To calculate it in one dimension, simply subtract the final position from the initial position. In symbols, if s is a position function with respect to time t, the displacement on the time interval [a,b] is: s(t) EX: Find the displacement between time 1 and 6. 2 t 1 2 3 4 5 6 -2 -4

  22. Average Velocity Two Points Needed It is the average rate of change or slope. s(t) EX: Find the average velocity between time 2 and 5. 2 t 1 2 3 4 5 6 -2 -4 The position function s can be used to find average velocity (speed) between two positions. Average velocity is the displacement divided by the total time. To calculate it between time a and time b:

  23. Instantaneous Velocity s(t) 2 t 1 2 3 4 5 6 -2 -4 One Point Needed EX: Graph the object’s velocity where it exists. v(t) Corner Slope = 0 2 Slope = -4 t 1 2 3 4 5 6 Slope = 2 -2 Slope = 4 -4 Slope = -2 The position function s can be used to find instantaneous velocity (often just referred to as velocity) at a position if it exists. Velocity is the instantaneous rate of change or the derivative of s at time t: Slope = 0

  24. Position, Velocity, … Units = Measure of length (ft, m, km, etc) The object is… Moving right/up when v(t) > 0 Moving left/down when v(t) < 0 Still or changing directions when v(t) = 0 Units = Distance/Time (mph, m/s, ft/hr, etc) Speed = absolute value of v(t) Position, Velocity, and Acceleration are related in the following manner: Position: Velocity:

  25. Example 1 s(t) 2 t 1 2 3 4 5 6 -2 -4 The position of a particle moving left and right with respect to an origin is graphed below. Complete the following: • Find the average velocity between time 1 and time 4. • Graph the particle’s velocity where it exists. • Describe the particles motion.

  26. Example 2 Sketch a graph of the function that describes the motion of a particle moving up and down with the following characteristics: The particle’s position is defined on [0,10] The particle’s velocity is only positive on (4,7) The average velocity between 0 and 10 is 0.

  27. Example 3 The derivative of the position function is the velocity function. The position of a particle is given by the equation where t is measured in seconds and s in meters. (a) Find the velocity at time t.

  28. Example 3 (continued) m/s m/s The position of a particle is given by the equation where t is measured in seconds and s in meters. (b) What is the velocity after 2 seconds? (c) What is the speed after 2 seconds?

  29. Example 3 (continued) The particle is at rest when the velocity is 0. After 1 second and 3 seconds The position of a particle is given by the equation where t is measured in seconds and s in meters. (d) When is the particle at rest?

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