MATH 181 Calculus I - Fall 2024 Lecture Notes PDF

MATH 181 Calculus I Fall 2024 Chapter 1. Functions and Limits Important Topics: Denition of function. The domain and range of a function. Piecewise dened Functions. Symmet...

MATH 181 Calculus I Fall 2024 Chapter 1. Functions and Limits Important Topics: Denition of function. The domain and range of a function. Piecewise dened Functions. Symmetry: odd and even functions. Increasing and decreasing. Composite functions. The limit of a function. One-sided limits. Continuity and discontinuity. Limits involving innity. Horizontal and vertical asymptotes. The Vertical Line Test: A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. Limit vs One-sided Limits: lim f (x) = L if and only x→a lim f (x) = L and lim f (x) = L. x→a− x→a+ Limit Laws: Suppose that is a constant and the limits x→a lim f (x) and lim g(x) exist. Then x→a (1) lim [f (x) + g(x)] = lim f (x) + lim g(x). x→a x→a x→a (2) lim [f (x) − g(x)] = lim f (x) − lim g(x). x→a x→a x→a (3) lim [cf (x)] = c lim f (x). x→a x→a (4) lim [f (x)g(x)] = lim f (x) · lim g(x). x→a x→a x→a f (x) lim f (x) (5) lim = x→a if lim f (x) 6= 0. x→a g(x) lim g(x) x→a x→a h in (6) lim [f (x)]n = lim f (x). x→a x→a   (7) lim n f (x) = n lim f (x). If n is even, we assume that lim f (x) > 0 p q x→a x→a x→a Comparison Theorem If f (x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then lim f (x) ≤ lim g(x). x→a x→a The Squeeze Theorem: If f (x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and lim f (x) = lim h(x) = L, x→a x→a then lim g(x) = L. x→a Dr. Hongtao Yang UNLV  Mathematical Sciences 1/8 MATH 181 Calculus I Fall 2024 Direct Substitution Property: Let f be a polynomial, a trigonometric, or a rational function. If a is a number in the domain of f , then lim f (x) = f (a). x→a Continuity: A function f is continuous at a number a if lim f (x) = f (a). x→a The Properties of Continuity: (1) If f and g are continuous at a and c is a constant, then f + g, f − g, cf , f g , and f /g (g(a) 6= 0) are also continuous at a. (2) Polynomials, rational functions, root functions, and trigonometric functions are continuous at every number in their domains. (3) If f is continuous at b and lim g(x) = b, then x→a   lim f (g(x)) = f (b) i.e., lim f (g(x)) = f lim g(x). x→a x→a x→a (4) If g is continuous at a and f is continuous at g(a), then the composite function f ◦g given by (f ◦g)(x) = f (g(x)) is continuous at a. The Intermediate Value Theorem: Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) 6= f (b). Then there exists a number c in (a, b) such that f (c) = N. Chapter 2. Derivatives Important Topics Tangent line and velocity. Average rate of change and instantaneous rate of change. Derivatives. Implicit dierentiation. Related rates. Linear approximations. Dierentials. Basic Dierentiation Formulas: If c is a constant and f and g are a dierentiable function, then d d (1) [cf (x)] = c f (x). dx dx d d d (2) [f (x) + g(x)] = f (x) + g(x). dx dx dx d d d (3) [f (x) − g(x)] = f (x) − g(x). dx dx dx d n The Power Rule: x = nxn−1. dx The Product Rule: If f and g are both dierentiable, then d d d [f (x)g(x)] = g(x) f (x) + f (x) g(x). dx dx dx Dr. Hongtao Yang UNLV  Mathematical Sciences 2/8 MATH 181 Calculus I Fall 2024 The Quotient Rule: If f and g are both dierentiable, then d d f (x) − f (x) dx   d f (x) g(x) dx g(x) = 2. dx g(x) [g(x)] The Chain Rule: If f and g are both dierentiable and F = f ◦ g is the composite function dened byF (x) = f (g(x)), then F is dierentiable and F is given by 0 F 0 (x) = f 0 (g(x)) · g 0 (x). The Derivatives of the Trigonometric Functions: d d d sin(x) = cos(x) cos(x) = − sin(x) tan(x) = sec2 (x) dx dx dx d d d cot(x) = − csc2 (x) sec(x) = sec(x) tan(x) csc(x) = − csc(x) cot(x) dx dx dx Linear Approximations: The linear or tangent line approximation of f at a is L(x) = f (a) + f 0 (a)(x − a). Dierentials: Let f be a dierentiable function. The dierential is an independent variable and the dierential dy is dened by dy = f 0 (x)dx. Chapter 3. Inverse Functions Important Topics Exponential functions and the natural exponential function. One-to-one function and its inverse function. Logarithmic Functions. Logarithmic dierentiation. Exponential growth and decay Dierential equations and their solutions. Inverse trigonometric functions. Hyperbolic functions. Indeterminate forms and L'hospital's rule. The Laws of Exponents ax (1) ax+y = ax ay (2) ax−y = (3) (ax )y = axy (4) (ab)x = ax bx ay Properties of Exponential Functions (1) If 0 < a < 1, then the exponential function y = ax is decreasing and lim ax = 0 and lim ax = ∞. x→∞ x→−∞ (2) If a > 1, then the exponential function y = ax is increasing and lim ax = ∞ and lim ax =. x→∞ x→−∞ So, the x-axis is a horizontal asympotes of the exponential functions. We have the following gure for the exponential functions. Dr. Hongtao Yang UNLV  Mathematical Sciences 3/8 MATH 181 Calculus I Fall 2024 Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once. Denition of Inverse Function Let f be a one-to-one function with domain A and range B. Then its inverse function f −1 has domain B and range A and is dened by f −1 (x) = y ⇐⇒ f (y) = x for any x in B (see the following diagram) So, domain of f −1 = range of f range of f −1 = domain of f Cancellation Equations f −1 (f (x)) = x for every x in A f f −1 (x) = x for every x in B
Find the Inverse Function of a One-to-one Function Step 1. Form the equation f (x) = y. Step 2. Solve the equation for x in terms of y Step 4. Interchange x and y to get the expression of f −1. The graph of f −1 : The graph of f −1 is obtained by reecting the graph of f about the line y = x. Continuity: If f is a one-to-one continuous function dened on an interval, then its inverse function f −1 is also continuous. Derivative formula: Let f be a one-to-one dierentiable function with inverse function f −1. If f −1 (a) = b (i.e., f (b) = a) and f 0 (b) 6= 0, then the inverse function is dierentiable at a and
0 1
0 1 f −1 (a) = , i.e., f −1 (a) =. f 0 (b) f 0 (f −1 (a)) Dr. Hongtao Yang UNLV  Mathematical Sciences 4/8 MATH 181 Calculus I Fall 2024 Derivatives of Logarithmic and Exponential Functions: d 1 d 1 d x d x (1) (loga (|x|)) = (2) ln(|x|) = (3) a = ln(a)ax (4) e = ln(a)ex. dx x ln(a) dx x dx dx Logarithmic Dierentiation. Exponential Growth and Decay. Derivatives of the Inverse Trigonometric Functions: d 1 (1) arcsin(x) = √ for all −1 < x < 1. dx 1 − x2 d 1 (2) arccos(x) = − √ for all −1 < x < 1. dx 1 − x2 d 1 (3) arctan(x) = for all −∞ < x < ∞. dx 1 + x2 Derivatives of Hyperbolic Functions and their Inverse Functions : d (1) sinh(x) = cosh(x) dx d (2) cosh(x) = sinh(x) dx √ (3) sinh−1 (x) = ln(x + 1 + x2 ) for all −∞ < x < ∞. d 1 (4) sinh−1 (x) = √ for all −∞ < x < ∞. dx 1 + x2 L'Hospital's Rule: Suppose f and g are dierentiable and g0 (x) 6= 0 near a (except possibly at a). Suppose that lim f (x) = 0 and lim g(x) = 0 x→a x→a or that lim f (x) = ±∞ and lim g(x) = ±∞. x→a x→a Then f (x) f 0 (x) lim = lim 0 x→a g(x) x→a g (x) provided that the limit on the right side exists (or is ∞ or −∞). Indeterminate dierences, products and powers Chapter 4. Applications of Dierentiation Absolute maximum and minimum values. Local maximum and minimum values. Critical number. Concave upward, concave downward and inection point. Antiderivative. Dr. Hongtao Yang UNLV  Mathematical Sciences 5/8 MATH 181 Calculus I Fall 2024 The Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute maxi- mum value f (c) and an absolute minimum value f (d) at some numbers c and d in [a, b]. Fermat's Theorem: If f has a local maximum or minimum at c, and if f 0 (c) exists, then f 0 (c) = 0. The Closed Interval Method: To nd the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: (1) Find the values of f at the critical numbers of f in (a, b). (2) Find f (a) and f (b). (3) The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Rolle's Theorem: Let f be a function that satises the following three hypotheses: (1) f is continuous on the closed interval [a, b]. (2) f is dierentiable on the open interval (a, b). (3) f (a) = f (b). Then there is a number c in (a, b) such that f (c) = 0. The Mean Value Theorem: Let f be a function that satises the following three hypotheses: (1) f is continuous on the closed interval [a, b]. (2) f is dierentiable on the open interval (a, b). Then there is a number c in (a, b) such that f (b) − f (a) f 0 (c) = , i.e., f (b) − f (a) = (b − a)f 0 (c). b−a Theorem 5: If f 0 (x) = 0 for all in an interval (a, b), then f is constant on (a, b). Corollary 7: If f 0 (x) = g0 (x) for all in an interval (a, b), then f − g is constant on (a, b) (i.e., f (x) = g(x) + c for a constant c). Increasing/Decreasing Test: (a) If f 0 (x) > 0 on an interval, then f is increasing on that interval. (b) If f 0 (x) < 0 on an interval, then f is decreasing on that interval. Concavity Test: (a) If f 00 (x) > 0 for all x in interval I , then the graph of f is concave upward on I. (b) If f 00 (x) < 0 for all x in interval I , then the graph of f is concave downward on I. The First Derivative Test for Local Extreme Values: Suppose that c is a critical number of a continuous function f. (a) If f 0 changes from positive to negative at c, then f has a local maximum at c. (b) If f 0 changes from negative to positive at c, then f has a local minimum at c. Dr. Hongtao Yang UNLV  Mathematical Sciences 6/8 MATH 181 Calculus I Fall 2024 (c) If f does not change sign at c (for example, if f 0 is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c. The Second Derivative Test for Local Extreme Values: Suppose f 00 is continuous near c. (a)If f 0 (c) = 0 and f 00 (c) > 0 , then f has a local minimum at c. (a)If f 0 (c) = 0 and f 00 (c) < 0, then f has a local maximum at c. First Derivative Test for Absolute Extreme Values. Suppose that is a critical number of a continuous function dened on an interval. (a) If f 0 (x) > 0 for all x < c and f 0 (x) < 0 for all x > c, then f (c) is the absolute maximum value of f. (b) If f 0 (x) < 0 for all x < c and f 0 (x) > 0 for all x > c, then f (c) is the absolute minimum value of f. Sketching a Curve Optimization Problems Newton's Method Chapter 5. Integrals Areas and distances. Partition of an interval, sample points, and the Riemann sum. The denite integral Integrand and limits of integration. The right and left rectangle rules and the midpoint rule. Indenite integral. Average value of a function. Theorem 3: If f is continuous on [a, b], or if f has only a nite number of jump discontinuities, then f is Z b integrable on [a, b]; that is, the denite integral f (x)dx exists. a Z a Properties of the Denite Integral: (1) f (x)dx = 0 a Z b Z a (2) f (x)dx = − f (x)dx a b Z b (3) c dx = c(b − a) a Z b Z a (4) cf (x)dx = c f (x)dx a b Z b Z b Z b (5) [f (x) + g(x)]dx = f (x)dx + g(x)dx a a a Z b Z b Z b (6) [f (x) − g(x)]dx = f (x)dx − g(x)dx a a a Z b Z c Z b (7) f (x)dx = f (x)dx + f (x)dx a a c Dr. Hongtao Yang UNLV  Mathematical Sciences 7/8 MATH 181 Calculus I Fall 2024 Comparison Properties of the Integral: Z b (8) If f (x) ≥ 0 for a ≤ x ≤ b, then f (x)dx ≥ 0. a Z b Z b (9) If f (x) ≥ g(x) for a ≤ x ≤ b, then f (x)dx ≥ g(x)dx. a a Z b (10) If m ≤ f (x) ≤ M for a ≤ x ≤ b, then m(b − a) ≤ f (x)dx ≤ M (b − a). a The Fundamental Theorem of Calculus: Suppose f is continuous on [a, b]. Z x (1) If g(x) = f (t)dt, then g 0 (x) = f (x). a Z b (2) f (x)dx = F (b) − F (a), where F is any antiderivative of f , that is, F 0 = f. a The Mean Value Theorem for Integral: If f is continuous on [a, b], then there exists a number c in [a, b] such that Z g 1 f (c) = fave = f (x)dx. b−a a The Substitution Rule for Indenite Integrals: If u = g(x) is a dierentiable function whose range is an interval I and f is continuous on I , then Z Z 0 f (g(x))g (x)dx = f (u)du. The Substitution Rule for Indenite Integrals: If g0 (x) is continuous on [a, b] and f is continuous on the range of u = g(x), then Z b Z g(b) 0 f (g(x))g (x)dx = f (u)du. a g(a) Integrals of Symmetric Functions: Suppose f is continuous on [a, b]. (1) If f is even (f (−x) = f (x)), then Z a Z a f (x)dx = 2 f (x)dx. −a 0 (2) If f is odd (f (−x) = −f (x)), then Z a f (x)dx = 0. −a Dr. Hongtao Yang UNLV  Mathematical Sciences 8/8