Slide 1 2.2 The derivative as a Function Slide 2 Recall: The derivative of a function number : at a fixed Definition (Derivative of ) For any number, the derivative of is Slide 3 Remark is a new function derived from called derivative. So, it is Domain of can be interpreted geometrically as the slope of the tangent line to the graph of at the point if it exists.
Slide 4 The graph of a function is given. Use it to sketch the graph of the derivative Slide 5 (Answer) The graph of a function is given. Use it to sketch the graph of the derivative Slide 6 The graph of a function is given. Use it to sketch the graph of the derivative
Slide 7 (Answer) The graph of a function is given. Use it to sketch the graph of the derivative Slide 8 a. If find a formula for b. Illustrate by comparing the graphs of and Slide 9 (Hint) a. b.
Slide 10 Let. a. Find the derivative of b. State the domain of Slide 11 (Hint) Slide 12 Find if Answer:
Slide 13 Notations D, differential operators ( meaning operation of differentiation ) Slide 14 Notation Leibniz notation Differentiation at Slide 15 Definition A function is differentiable at a if exists. is differentiable on an open interval [or (a, ), (,a) or (, )] if is differentiable at every number in the interval.
Slide 16 Where is the function Hint: Divide the cases into when. differentiable? and Slide 17 Theorem If is differentiable at, then is continuous at. Remark: The converse of the above theorem is not true. Slide 18 How can a function fail to be differentiable? Vertical tangent line ( Corner
Slide 19 How can a function fail to be differentiable? Discontinuity Slide 20 Higher Derivatives ( ) Slide 21 If find and Remark: All higher order derivatives whose order is greater than the order of the given polynomial vanishes.
Slide 22 2.3. Differentiation Formulas Slide 23 Derivative of a constant function : number Hint: Verify straightforwardly. Verify geometrically by relating the definition with the graph. Slide 24 Derivative of Power Functions
Slide 25 The Constant Multiple Rule If c is a constant and f is a differentiable function, then Slide 26 1. Differentiate 2. 3. 4. Slide 27 The Sum and difference Rule If f and g are both differentiable, then
Slide 28 Slide 29 Find the points on the curve where the tangent line is horizontal. Answer: Slide 30 The equation of motion of a particle is where is measured in centimeters and in seconds. Find the acceleration as a function of time. What is the acceleration after 2 seconds?
Slide 31 The Product Rule If f and g are both differentiable, then The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. Slide 32 Differentiate Slide 33 If and and it is known that, find
Slide 34 The Quotient Rule If and are differentiable, then The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Slide 35 Let. Find the derivative Slide 36 If Power Rule for negative power function is a positive integer, then
Slide 37 1. Differentiate 2. Slide 38 The Power Rule( General Version ) If n is any real number, then ( Slide 39 1. Differentiate 2. 3.
Slide 40 Find equations of the tangent line and normal line to the curve at the point Answer: Tangent: Normal: Slide 41 At what points on the hyperbola tangent line parallel to the line 3 is the Answer: Slide 42 2.4. Derivatives of Trigonometric Functions
Slide 43 Properties of Trigonometric function Trigonometric functions, such as sine, cosine, tangent, cotangent, cosecant and secant functions are defined for all real numbers x. For example, f (x) = sin x. Here, real numbers are radian measure, rather than degree. All of the trigonometric functions are continuous at every number in their domains. Slide 44 Derivative of sine function is cosine function. (Graphical and intuitive explanation) Slide 45 Two important formula
Slide 46 Find. Hint: Use 1. 2. Slide 47 Calculate. Hint: Use 1. 2. Slide 48 Derivatives of Trigonometric Functions
Slide 49 Differentiate y = x 2 sin x. Slide 50 Use of Trigonometric Functions Trigonometric functions are often used in modeling real-world phenomena. In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions. In the following example we discuss an instance of simple harmonic motion. Slide 51 An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0. (See Figure and note that the downward direction is positive.) Its position at time t is s = f(t) = 4 cos t Find the velocity and acceleration at time t and use them to analyze the motion of the object.
Slide 52 ( Hint ) The velocity and acceleration are and. The object oscillates from the lowest point (s = 4 cm) to the highest point (s = 4 cm). The period of the oscillation is 2, the period of cos t. Slide 53 ( Further ) The speed is v = 4 sin t, which is greatest when sin t = 1, that is, when cos t = 0. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t = 0, that is, at the high and low points. The acceleration a = 4 cos t = 0 when s = 0. It has greatest magnitude at the high and low points. See the graphs in the Figure. Slide 54 2.5. The Chain Rule
Slide 55 How can we differentiate? We know the derivative formula for and. We may view F(x) as a composite function: Then, Use the Chain rule. Slide 56 Chain Rule Slide 57 Find if. Hint: 1. Analyze the given function to write that as a composite function. 2. Let the inner function as 3. Write the outer function as a function of u. That is, 4. Apply the Chain rule:. 5. Substitute for.
Slide 58 Chain Rule for Sine and Cosine Functions If y = sin u, where u is a differentiable function of x, then. If y = cos u, where u is a differentiable function of x, then. Slide 59 Differentiate 1. 2. Slide 60 The Chain Rule for the Power Function This rule is applied to the functions such as, etc.
Slide 61 Differentiate 1. 2. Slide 62 2.6. Implicit Differentiation Slide 63 How can we differentiate a smooth curve such as x 2 + y 2 = 25, x 3 + y 3 = 6xy? The collection of the points which satisfies the equation on the Cartesian plane is not the graph of a function because that doesn t pass the vertical line test. However, we still can consider the concept of slope of the graph locally at most points of the graph. Implicit differentiation makes the calculation possible.
Slide 64 If x 2 + y 2 = 25, find. Hint: 1. Differentiate as usual if the term is associated with only x. 2. Use the chain rule when the term contains y. Slide 65 Find an equation of the tangent to the circle x 2 + y 2 = 25 at the point (3, 4). Answer: 3x + 4y = 25 Slide 66 a. Find if b. Find the tangent to at the point (3,3). c. At what point in the first quadrant is the tangent line horizontal?
Slide 67 Answer a. b. c. when (0,0) is a double point which has two tangent lines. The tangent is horizontal at. (Remark: and are exponents.) Slide 68 Find if Answer: Slide 69 Linear Approximation and Differentials
Slide 70 Notion of derivative Local figure of a curve Slide 71 By zooming in toward a point on the graph of a differentiable function, the graph looks more and more like its tangent line. That is, f(x) f(a) + f (a)(x a) near the point Slide 72 Definitions f(x) f(a) + f (a)(x a) is called the linear approximation or tangent line approximation of f at a. The linear function whose graph is this tangent line, that is, L(x) = f(a) + f (a)(x a) is called the linearization of f at a.
Slide 73 Find the linearization of the function f(x) = at x = 1 and use it to approximate the numbers and. Are these approximations overestimates or underestimates? Hint: Slide 74 Answer: Overestimate 2.0125 Slide 75 Linear Approximations and Differentials The tangent line approximation gives good estimates when x is close to 1 but the accuracy of the approximation deteriorates when x is farther away from 1.
Slide 76 Numerical Evidence Slide 77 Differentials If y = f(x), where f is a differentiable function, then the differential dx is an independent variable; that is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation dy = f (x) dx Thus, dy is a dependent variable; it depends on the values of x and dx. Slide 78 The geometric meaning of differentials
Slide 79 Compare the values of y and dy if y = f(x) = x 3 + x 2 2x + 1 and x changes (a) from 2 to 2.05 and (b) from 2 to 2.01. Hint and Answer: (a) f(2)=9, f(2.05)=9.717625, y= 0.717625 dy= (3x 2 + 2x 2) dx, dy=0.7. (b) f(2.01) = 9.140701, y= 0.140701, dy = 0.14