Derivative of a Constant Multiple of a Function Theorem: If f is a differentiable function and if c is a constant, then

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1 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 1 Review of the Limit Definition of the Derivative Write the it efinition of the erivative function: f () Derivative of a Constant Multiple of a Function Theorem: If f is a ifferentiable function an if c is a constant, then c f () We also use the notation f () c Eercise 1: Prove this Theorem using the efinition of the erivative. c f () h0 h0 Eercise : Illustrate the Theorem above with a geometric eample. y f() Estimate the erivative of f at -. Estimate the erivative of f at 3. y g() 4f() Estimate the erivative of g at -. Estimate the erivative of g at 3.

2 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete Derivative of the Sum or Difference of Functions Theorem: If f an g are ifferentiable functions, then g( ) ( ) g( ) f ( ) f Eercise 3: Prove the Difference Rule using the efinition of the erivative. (The proof for the Sum Rule is foun on page 130 in the tet.) f ( ) g( ) h0 h0 Derivative of a Power Function (The Power Rule) Def: A power function is a function of the form, where Theorem: The erivative function of f n ( ) is Note: The proof an/or hints for a proof of the Power Rule for various conitions on n can be foun on pages 17, 143, an 173 in the tet. Derivative of y c an of y The erivative of the constant function f() c is f () The erivative of the function g() is g () That is, (c) That is, () Use the Power Rule to show this fact. f() c Use the Power Rule to show this fact. g() f () Use the it efinition to prove this fact. g () Use the it efinition to prove this fact.

3 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 3 Eercise 4: Use the Power Rule to etermine each erivative function. (i) y 13 (ii) y (iii) y (iv) 4 y 4 1 (v) y 3 (vi) y 3 (vii) y π (viii) y ln() Derivative Functions of Polynomials an of Some Other Functions We can now etermine the erivative function of a function epresse (or epressable) as the sum or ifference of power functions, incluing polynomials. (Note that only the function in (i) below is a polynomial.) Eercise 5: Determine the erivative function. 3 (i) y (ii) y (iii) y (iv) 4 y 3 1

4 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 4 Eercise 6: What is erivative function of f() sin()? To answer this question, consier the graph of y sin() on the interval 0 rawn below. As we i in Hanout 3.1, raw the tangent line at various points, estimate the slope, an sketch f (). 0 3 f () Draw f () below. Conclusion: If f() sin(), then f () Eercise 7: What is erivative function of g() cos()? To answer this question, consier the graph of y cos() on the interval 0 rawn below. As we i in Hanout 3.1, raw the tangent line at various points, estimate the slope, an sketch g (). 0 3 g () Draw g () below. Conclusion: If g() cos(), then g ()

5 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 5 The Derivative of an Eponential Function Eercise 8: Consier f(). Write the it efinition of each of the following, an use the table feature of the TI to estimate the value of each it to four ecimal places. f (0) h0 f (0 h) h f (0) h0 f (1) h0 f () f (0) f (1) f () So, f () 0 f (0) 1 f (1) f () f () We recognize the ecimal as an approimation of We conclue that if f(), then f () Note that the erivative function of f() is merely a constant multiple of f(). The constant multiple, in this case, is If f() 3, then f (). Note that the erivative function of f() 3 is merely a constant multiple of f() 3. The constant multiple, in this case, is Recall: If one function is a constant multiple times another function, then its graph is a vertical stretch or shrink (epening on whether the constant multiple is greater than 1 or less than 1) of the graph of the other function.

6 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 6 Question: Is there an eponential function such that the constant multiple in the formula for the erivative function is 1? That is to say, is there an eponential function that is equal to its erivative function? Eercise 9: Let f() a. Write the it efinition of the erivative function, an compute the value of a for which f() f (). f () h0 f ( h) h f ( ) h0 Derivative Function of an Eponential Function y a (where a > 0) a Special case worth nothing: e (see p.158 in the tet)

7 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 7 Eercise 10: Determine the erivative function of each of the following functions. (i) y 4 (ii) y 5 (iii) y -6π + 3e (iv) y 3 (v) y (ln()) (vi) y (vii) y (e) (viii) y e 3

8 Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 8 Average Velocity an Instantaneous Velocity of a Rocket Eercise 11: A rocket is launche vertically from a submarine 00 feet below the surface of the ocean with an initial velocity of 1500 feet per secon. (For simplicity, we will ignore water friction an air friction. In reality, neither can be ignore.) The following table an graph represent y the height in feet of the rocket above the level of the water as a function of time in secons since the beginning of the launch. (The graph oes not inicate the path of the rocket.) The equation which relates y an is y Note that the -ais is scale in increments of 10, an the y-ais is scale in increments of (i) Using the table, compute the average velocity of the rocket between 0 sec. an 50 sec. Draw the corresponing secant line. (ii) Compute the average velocity of the rocket between 0 sec. an 5 sec. Draw the corresponing secant line. (iii) The instantaneous velocity of the rocket at 0 secons is the slope of the tangent line to the graph at 0. Compute this value eactly (using erivative rules, not by estimating a slope geometrically.) (iv) What is the velocity ( velocity means instantaneous velocity unless otherwise qualifie) of the rocket at the moment in its escent that it is 10,000 feet above the surface of the water?