Calculus with Dr. Bob III: Log, Exponential and Inverse Trig Functions

Video Lectures

Displaying all 48 video lectures.
 Lecture 1 Play Video Definition of ln(x) Using IntegrationCalculus: We define the natural logarithm as an antiderivative of 1/x using the Second Fundamental Theorem of Calculus. We investigate the graph and explain how to estimate values of ln(x). ADDED: Some methods for approximating e using calculus. Lecture 2 Play Video Properties of ln(x) Using IntegrationCalculus: We derive two key properties for ln(x) as a definite integral: the exponent rule and the product rule. We provide several methods for each property, including u-substitution and uniqueness of antiderivatives up to a constant. Lecture 3 Play Video End Behavior of ln(x)Calculus: Using the exponent rule for natural logarithm, we verify the end behavior of the graph of ln(x). That is, we show that the limit as x goes to infinity of ln(x) is positive infinity and that the limit as x goes to 0 from the right is negative infinity. We interpret the result as an area. Lecture 4 Play Video Derivative of ln(x) as a SlopeCalculus: With the properties developed so far, we verify the derivative of ln(x) by using the definition as the slope of a tangent line. A special limit is required, which we verify at a few points. Lecture 5 Play Video The Chain Rule for ln(x) and ln|x|Calculus: We consider derivatives for functions of the form ln(g(x)) and ln|g(x)|. As preparation, we note some details concerning the domain of ln(x). Example include y(x) = ln|x^2-4x| and y(x) = ln|ln(x)|. Lecture 6 Play Video Derivative of f(x) = ln(ln(x^2 + 1))Worked problem in calculus. The derivative of f(x) = ln(ln(x^2 + 1)) is calculated using the chain rule twice. Lecture 7 Play Video Graph of f(x) = 3ln(x-2) + 4Worked problem in calculus. The graph of 3ln(x-2) + 4 is derived from the graph of ln(x). Lecture 8 Play Video Graph of f(x) = ln(x^2 - x)Worked problem in calculus. Sketch the graph of ln(x^2-x). Regions of increase/decrease and concavity are determined. Lecture 9 Play Video Graph of f(x) = ln|x^2-x|Calculus: Using the chain rule for ln|x|, we sketch the graph of f(x) = ln|x^2-x|. Steps include finding the domain, regions of increasing-decreasing and concavity, and end behavior. Lecture 10 Play Video Implicit Differentiation with ln(x)Worked problem in calculus. Find the tangent line to the curve xy + ln(xy^2) = 1 at the point (1, 1). Implicit differentiation is used to find the slope of the tangent line. Lecture 11 Play Video Logarithmic DifferentiationWorked problem in calculus. Find the derivative f'(x) of the function f(x) = (x+1)^(1/3)/[x^2(x+2)^(1/5)] using logarithmic differentiation. Logarithmic differentiation is a useful alternative to the quotient rule for derivatives. Lecture 12 Play Video Antiderivative of ln(x^4)/xCalculus: The antiderivative of ln(x^4)/x is computed in two ways, both using integration by substitution. The main concept is that the derivative of ln|x| is 1/x. Care must be taken when simplifying with the exponent rule for logarithm. Lecture 13 Play Video Antiderivative involving 1/x 1Worked problem in calculus. The antiderivative of f(x) = (x^3-2x^2+x+1)/(x+2) is computed by two methods. The first uses synthetic division. The second uses the substitution u = x+2. In both cases, a key step is that int 1/x dx = ln|x| + C Lecture 14 Play Video Antiderivative involving 1/x 2Worked problem in calculus. We calculate the indefinite integrals of the functions (a) f(x) = 1/[x^(1/3) (1+x^(2/3))] and (b) f(x) = x(x-3)/(x + 2)^3. A key step uses that int 1/x = ln|x| + c. Lecture 15 Play Video Trig Antiderivatives involving 1/xWorked problem in calculus. Find the indefinite integral of (a) f(x) = 2x sec(1+x^2) and (b) f(x) = tan(ln(x))/x. A key step is that int 1/x dx = ln|x| + c. Lecture 16 Play Video Area under f(x) = ln(x)/xWorked problem in calculus. The area under the curve f(x) = ln(x)/x between x=1 and x=e above the x-axis is calculated. A key step is that int 1/x dx = ln|x| + c. Lecture 17 Play Video Inverse FunctionsWorked problem in calculus. Determine if the following functions have inverses. If so, find the inverse function, check, and sketch. (a) f(x) = (x-3)^3 + 4, (b) f(x) = |x-2| + |x-1|, and (c) f(x) = 2sqrt(x - 3) + 1. Lecture 18 Play Video Inverse Functions: ln(x) and exp(x)Calculus: Because ln(x) is a one-one function, we consider its inverse function exp(x). We interpret the usual properties for inverse functions with these functions. Lecture 19 Play Video Graph of f(x) = 2 - e^{2x-1}Worked problem in calculus. The graph of f(x) = 2- e^(2x-1) is derived from the graph of g(x) = e^x. Lecture 20 Play Video Inverse Function for f(x) = 2 e^{x-2}Worked problem in calculus. For the function f(x) = 2 e^(x-2), the inverse is found and checked. Then the derivative rule for inverse functions is verified. Lecture 21 Play Video No Inverse Function for f(x)=ln(x)-exp(x)Calculus: Let f(x) = ln(x) - exp(x). We show that f(x) has no inverse function by considering 1) end behavior and 2) regions of increasing an decreasing. By restricting the domain, we find an inverse function and approximate f^{-1}(1) and f^{-1}(-3). Lecture 22 Play Video Derivative of an Inverse FunctionWorked problem in calculus. Find the inverse function, check, and verify the derivative rule for the functions (a) f(x) = 3x-2, and (b) g(x) = sqrt(x-3). Lecture 23 Play Video Properties of Derivative of exp(x)Calculus: We show that the derivative of exp(x) is exp(x). We verify derivative properties of the graph of exp(x), show consistency with the slope definition of derivative, and note how the chain rule applies. Lecture 24 Play Video Derivatives with e^xWorked problem in calculus. The derivative is found for the following functions: (a) f(x) = x^2 e^{-x+ln(x)}, (b) g(x) = ln(1+e^{x^2}), and (c) h(x) = int_0^ln(x) sqrt(1+e^t) dt. Part (c) is an application of the Second Fundamental Theorem of Calculus. Lecture 25 Play Video Tangent Line to x^2 e^x at x=1Worked problem in calculus. The formula for the tangent line to f(x) = x^2 e^{-x} at x=1 is derived. The line is used to approximate f(1.1). Lecture 26 Play Video Graph of f(x) = x^2 e^{-x}Worked problem in calculus. The graph of f(x) = x^2 e^{-x} is derived from the first and second derivatives. Lecture 27 Play Video Graph of Normal DistributionCalculus: As an application of the chain rule with exp(x), we sketch the function f(x) = exp(-(x-m)^2/(2s^2)), a multiple of a normal distribution. Using the first and second derivatives, we note critical and inflection points, and regions of increasing, decreasing, and concavity. Lecture 28 Play Video Implicit Differentiation with e^xCalculus: Consider the curve defined by exp(xy) + x^2 + y^2 = 5. Use implicit differentiation to find the tangent line to the curve at (2, 0). Then use the tangent line to approximate the point near (2, 0) with x = 2.01. Lecture 29 Play Video Indefinite Integrals with e^xWorked problem in calculus. Evaluate the indefinite integrals of (a) e^{-3x+4}, (b) (e^x - 2)(e^x -3)/e^x and (c) e^x/ (1 + 2e^x). Lecture 30 Play Video Definite Integral of exp(-3x+2) (HD Version)Calculus: Compute the definite integral of exp(-3x+2) over the interval [0, ln(2)]. We apply integration by substitution to solve. Lecture 31 Play Video Definite Integral of (e^x - 1)(e^x - 2)/e^2x (HD Version)Calculus: Compute the definite integral of (e^x - 1)(e^x - 2)/e^2x over the interval [ln(2), ln(3)]. We use integration by substitution to solve. Lecture 32 Play Video Graphing with 2^xWorked problem in calculus. Graph the function f(x) = 3.2^(x-1) - 4. Lecture 33 Play Video Equations with LogarithmsWorked problem in calculus. The equations log_6(x-5) + log_6(x) = 1 and log_b(9) - 3log_b(3) = 1/2 are solved. Lecture 34 Play Video Derivatives of log_b(x) and b^xWorked problem in calculus. Find the derivative of f(x)= x.2^(3x+1) and g(x) = log_2(x^2-2x-3/x). Lecture 35 Play Video Tangent Lines to log_b(x) and b^xWorked problem in calculus. Compute the tangent lines to the following functions at x=1 and approximate at x=1.1. (a) f(x) = 3^(x^2-x), (b) f(x) = log_2(x+1). Lecture 36 Play Video Logarithmic DifferentiationWorked problem in calculus. Logarithmic differentiation is used to compute the derivative of f(x) = (x^2-x)^(2x+1). Lecture 37 Play Video Indefinite Integrals with b^x and log_b(x)Worked problem in calculus. Find the indefinite integrals (a) (2x-1) int 2^(x^2-x) dx and (b) int log_2(x)/x dx. Lecture 38 Play Video Definite Integrals with b^x and log_b(x)Worked problem in calculus. The following definite integrals are computed: (a) int_1^2 2^(3x-1) dx, and (b) int_1^2 [log_2(x)]^3/x dx. Lecture 39 Play Video Compound InterestWorked problem in calculus. Given an initial investment of \$1000 and a nominal annual interest rate of 5%, compute the amount accrued after 10 years by compounding annually, monthly, daily, and continuously. Find the time to double the initial investment if compounding is annual or continuous. Lecture 40 Play Video Evaluating Inverse Trig Functions 1Worked problem in calculus. We evaluate the inverse trig functions: (a) sin^{-1}(sqrt(3)/2), (b) cos^{-1}(3), (c) tan^{-1}(sqrt(3)/3), and (d) sec^{-1}(sqrt(2)). Lecture 41 Play Video Evaluating Inverse Trig Expressions 2Worked problem in calculus. Evaluate (a) cos(sin^{-1}(3/5)), (b) tan(sin^{-1}(-3/5)), and (c) csc(sec^{-1}(sqrt(2)) Lecture 42 Play Video Evaluating Inverse Trig Expressions 3Worked problem in calculus. Find the algebraic expressions for (a) cos(sin^{-1}(x/(x+1)), (b) sin(tan^{-1}(x)), and (c) csc(sec^{-1} (x)). Lecture 43 Play Video Derivatives of Inverse Trig FunctionsWorked problem in calculus. Find the derivatives of (a) g(x) = xsin^{-1}(x^2+x), (b) h(x) = tan^{-1}(ln(x)), and (c) s(x) = sec^{-1}(e^2x). Lecture 44 Play Video Graph of f(x) = tan^{-1}(1-x^2)Worked problem in calculus. Assuming the graph has a horizontal asymptote at y = pi/2, sketch the graph of f(x) = tan^{-1}(1-x^2) using the first and second derivatives. We make use of the quadratic equation for ax^4 + bx^2 + c with y=x^2. Lecture 45 Play Video Tangent Line to f(x) = xsin^{-1}(x/2)Worked problem in calculus. We compute the equation of the tangent line to f(x) = xsin^{-1}(x/2) at x=1. Using the tangent line, we approximate f(1.1). Lecture 46 Play Video Integrals with Inverse Trig Functions 1Worked problem in calculus. The indefinite integrals int 1/[4 + (x+2)^2] dx and int dx/sqrt(-4x-x^2) are calculated. The first integral uses the rule int du/(1+u^2) = tan^(-1) (u) + C. The second integral uses completing the square and the rule int du/sqrt(1-u^2). Lecture 47 Play Video Integrals with Inverse Trig Functions 2Worked problem in calculus. The definite integrals int_(-3)^0 dx/(x^2+6x+18), int_(-3)^0 (2x+6)dx/(x^2+6x+18), and int_(-3)^0 (2x+8)dx/(x^2+6x+18) are computed. The different approaches to each integral are noted. Lecture 48 Play Video Integrals with Inverse Trig Functions 3Worked problem in calculus. We calculate the definite integrals int_1^2 dx/sqrt(4-x^2) and int_0^(sqrt(3)/2) sin^-1 (x) dx/sqrt(1-x^2). The first integral uses the rule int dx/sqrt(1-x^2) = sin^-1(x) + C. The second rule is the special u-substitution when we have f(x)f'(x) as integrand.