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p105 Section 2.2: Basic Differentiation Rules and Rates of Change Find the derivative of a function using the Constant Rule Find the derivative of a function using the Power Rule Find the derivative of a function using the Constant Multiple Rule Find the derivative of a function using the Sum and Difference Rule Find the derivative of the sine function and of the cosine function Use derivatives to find rates of change Theorem 2.2: The Constant Rule ** Since the slope of a horizontal line is 0, the derivative of a constant function is 0 Theorem 2.3: The Power Rule ** If a variable is under a radical or in the denominator you will usually (not always though) rewrite the radical as a rational (fraction) exponent and the denominator as a negative exponent 5
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Theorem 2.4: The Constant Multiple Rule If f is a differentiable function and c is a real number, then cf is also differentiable and This says that is a differentiable function is multiplied by a constant, then its derivative is multiplied by the same constant. ** The Constant Multiple Rule and the Power Rule can be combined into one rule. 7
Theorem 2.5: The Sum & Difference Rules The sum (or difference) of two differentiable functions is differentiable and is the sum (or difference) of their derivatives. That means you can take the derivative of each term (especially helpful for polynomials) 8
Theorem 2.6: Derivatives of Sine and Cosine Functions 9
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Rates of Change The derivative can be used to determine the rate of change of one variable with respect to another. A common use is to describe the motion of a object moving in a straight line. If, over a period of time Δt, the object changes its position by the amount Δs = s(t + Δt) s(t), then by the familiar formula the average velocity is: Vavg = total displacement = total time Change in Change in ** When the derivative is negative, the velocity is moving in a negative direction and when the derivative is positive, the velocity is moving in a positive direction ** Slope of tangent line = derivative = velocity Instantaneous Velocity (that's simply the velocity), if s = s(t) is the position function for an object moving along a straight line, the velocity of the object at time t is ** The velocity function is the derivative of the position function Velocity can be positive, negative or zero ** The speed of an object is the absolute value f its velocity (it can't be negative) The position of a free falling object (without considering air resistance) under the influence of gravity can be represented by the following position equation: s 0 = initial height v 0 = initial velocity g = acceleration due to gravity = 32 feet per second = 9.8 meters per second 12
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Example 10: At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by: where s is measured in feet and t is measured in seconds. a. When does the diver hit the water? b. What is the diver's velocity at impact? 14
92. A ball is thrown straight down from the top of a 220 foot building with an initial velocity of 22 feet per second. What is the velocity after 3 seconds? What is its velocity after falling 108 feet? Position Function 94. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 6.8 seconds after the stone is dropped? Position Function: 15
96. Use the given position graph (which represents the distance in miles that a person drives during a 10 minute trip to work), make a sketch of the corresponding velocity function. Distance (in miles) y 7 6 5 4 3 2 1 (6, 5) (8, 5) (10, 6) (0, x 0) 2 0 2 4 6 8 10 12 1 Time (in minutes) Velocity (in mph) 100 y 10 909 808 707 606 505 404 303 202 101 0 1 2 3 4 5 6 7 8 9 10 Time (in minutes) x 98. The graph shown below gives the graph of a velocity function, which represents the velocity in miles per hour during a 10 minute drive to work. Make a sketch of the corresponding position function. Velocity (in mph) y 10 100 909 808 707 606 505 404 303 202 101 x Distance (in miles) y 10 9 8 7 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 Time (in minutes) 0 1 2 3 4 5 6 7 8 9 10 Time (in minutes) 16
102. The volume of a cube with sides of length s is given by V = s 3. Find the rate of change of the volume with respect to s when s = 4 centimeters. 17