CALCULUS II MATH 2414 Dr. Hyunju Ban
Introduction Syllabus Chapter 5.1 5.4
Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of Integration (1.5 weeks) Chap 8: Applications of Integration (2 weeks) Chap 9: Infinite Series (2.5 week) Chap 10: Conics, Parametric Equations, and Polar Coordinates (1.5 weeks)
Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions 5.1 The Natural Logarithmic Function: Differentiation 5.2 The Natural Logarithmic Function: Integration 5.3 Inverse Functions 5.4 Exponential Functions: Differentiation and Integration 5.5 Bases Other than e and Applications 5.6 Indeterminate Forms and L Hopital s Rule 5.7 Inverse Trigonometric Functions: Differentiation 5.8 Inverse Trigonometric Functions: Integration 5.9 Hyperbolic Functions
5.1. The Natural Logarithmic Function: Differentiation The natural logarithmic function is defined by ln x = 1 x 1 t dt, x > 0. The domain of the natural logarithmic function is the set of all positive real numbers. Properties. The natural logarithmic function has the following properties. The domain is (0, ) and the range is,. The function is continuous, increasing, and one-to-one. The graph is concave downward.
5.1. The Natural Logarithmic Function: Differentiation The natural logarithmic function is defined by ln x = 1 x 1 t dt, x > 0. The domain of the natural logarithmic function is the set of all positive real numbers. Properties. If a and b are positive numbers and n is rational, then the following properties are true. ln 1 = 0 ln ab = ln a + ln b ln a n = n ln a ln a b = ln a ln b
5.1. The Natural Logarithmic Function: Differentiation The natural logarithmic function is defined by ln x = The letter e denotes the positive real number such that ln e = 1 e 1 t dt = 1. Note that e is an irrational number and e 2.718. 1 x 1 t dt, x > 0. The domain of the natural logarithmic function is the set of all positive real numbers.
5.1. The Natural Logarithmic Function: Differentiation The natural logarithmic function is defined by Derivatives of the Natural Logarithmic Function. Let u be a differentiable function of x. Then d dx ln x = 1 x, x > 0 d ln x = 1 dx ln u = 1 u d ln u dx = x 1 t u u. dt, x > 0. The domain of the natural logarithmic function is the set of all positive real numbers. du dx = u u, u > 0
5.2. The Natural Logarithmic Function: Integration The natural logarithmic function is defined by ln x = 1 x 1 t dt, x > 0. The domain of the natural logarithmic function is the set of all positive real numbers. Log Rule for Integration Let u be a differentiable function of x. Then 1 x dx = ln x + C, 1 u du = ln u + C, u u dx = ln u + C.
5.2. The Natural Logarithmic Function: Integration ln x = 1 x 1 t dt x > 0, u u dx = ln u + C. Integrals of the six basic trigonometric functions sin u = cos u + C, cos u du = sin u + C tan u du = ln cos u + C, cot u du = ln sin u + C sec u du = ln sec u + tan u + C csc u du = ln csc u + cot u + C
5.3. Inverse Functions A function g is the inverse function of the function f when and f(g x ) = x for each x in the domain g g(f x ) = x for each x in the domain f. The function g is denoted by f 1 (read f inverse ) Existence of an Inverse Function A function has an inverse function if and only if it is one-to-one. If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.
5.3. Inverse Functions A function g is the inverse function of the function f when and f(g x ) = x for each x in the domain g g(f x ) = x for each x in the domain f. The function g is denoted by f 1 (read f inverse ) Properties If g is the inverse function of f, then f is the inverse function of g. The domain of f 1 is equal to the range of f, and the range of f 1 is equal to the domain of f. A function need not have an inverse function, but when it does, the inverse function is unique. The graph of f and the graph of f 1 are symmetric about the line y = x. In other words, the graph of f contains the point (a, b) if and only if the graph of f 1 contains the point (b, a).
5.3. Inverse Functions A function g is the inverse function of the function f when f(g x ) = x for each x in the domain g and g(f x ) = x for each x in the domain f. The function g is denoted by f 1 (read f inverse ) How to Find? 1. Determine whether the function has an inverse function. 2. Solve for x as a function of y: x = g y = f 1 y 3. Interchange x and y. The resulting equation is y = f 1 x. 4. Define the domain of f 1 as the range of f. 5. Verify that f f 1 x = x and f 1 f x = x.
5.3. Inverse Functions A function g is the inverse function of the function f when and f(g x ) = x for each x in the domain g g(f x ) = x for each x in the domain f. The function g is denoted by f 1 (read f inverse ) Continuity and Differentiability of Inverse Functions Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true. 1. If f is continuous on its domain, then f 1 is continuous on its domain. 2. If f is increasing on its domain, then f 1 is increasing on its domain. 3. If f is decreasing on its domain, then f 1 is decreasing on its domain. 4. If f is differentiable on its domain containing c and f c 0, then f 1 is differentiable at f(c).
5.3. Inverse Functions A function g is the inverse function of the function f when and f(g x ) = x for each x in the domain g g(f x ) = x for each x in the domain f. The function g is denoted by f 1 (read f inverse ) The Derivative of an Inverse Function Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f g x 0. Moreover, g x = 1 f g(x), f g x 0.
5.4. Exponential Functions: Differentiation and Integration Definition The inverse function of the natural logarithmic function f x = ln x is called the natural exponential function and is denoted by That is, Note that f 1 x = e x. y = e x if and only if x = ln y. ln(e x ) = x and e ln x = x. Operations with Exponential Functions Let a and b be any real numbers. e a e b = e a+b e a = ea b eb
5.4. Exponential Functions: Differentiation and Integration Definition The inverse function of the natural logarithmic function f x = ln x is called the natural exponential function and is denoted by That is, Note that f 1 x = e x. y = e x if and only if x = ln y. ln(e x ) = x and e ln x = x. Properties of the Natural Exponential Function The domain of f x = e x is, and the range is (0, ) The function f x = e x is continuous, increasing, and one-to-one on its entire domain. The graph of f x = e x is concave upward on its entire domain. lim x ex = 0 and lim x e x =.
5.4. Exponential Functions: Differentiation and Integration Definition The inverse function of the natural logarithmic function f x = ln x is called the natural exponential function and is denoted by That is, Note that f 1 x = e x. y = e x if and only if x = ln y. ln(e x ) = x and e ln x = x. Derivatives of the Natural Exponential Function Let u be a differential function of x. Then d dx ex = e x d dx eu = e u du dx
5.4. Exponential Functions: Differentiation and Integration Definition The inverse function of the natural logarithmic function f x = ln x is called the natural exponential function and is denoted by That is, Note that f 1 x = e x. y = e x if and only if x = ln y. ln(e x ) = x and e ln x = x. Integration Rules for the Natural Exponential Function Let u be a differential function of x. Then e x dx = e x + C e u du = e u + C