Business Calculus by Kaiju Mathematics: Become a Monster

Explore limits using left and right approaches, numerical and graphical methods, and algebraic factoring, showing the limit as x approaches 3 is -1 and as x approaches 4 is 8.

Explore evaluating left and right limits for piecewise and rational functions, determine when limits exist or diverge, and distinguish holes from asymptotes on graphs.

Explore algebraic limits and continuity by applying limit laws to evaluate polynomials and roots, showing when to plug in values and respect domain constraints.

Use factoring to evaluate limits, check domain to avoid zero denominators, and compute limits as x approaches 6 and x approaches 3 with difference of squares and trinomial factoring.

Multiply by 1/x^2 to evaluate the limit as x goes to infinity, canceling terms and yielding a limit of 2.

Check that f(a) exists and the limit as x approaches a exists and matches f(a) to confirm continuity. Identify infinite, removable, and jump discontinuities, including a hole at x=5.

Explore average rate of change and the difference quotient, using f(x2)−f(x1) over (x2−x1) to find slope between two points. See secant lines converge to a tangent line and begin derivatives.

Compute the average rate of change of f(x)=x^2+2 between x=1 and x=3, and between x=1 and x=2, using secant lines to approach the tangent line and its slope.

Explore difference quotient as a bridge to derivatives by evaluating (f(x+h) - f(x)) / h with h not equal to zero and substituting x+h for x in x^2 - x.

Compute the simplified difference quotient for f(x)=x^3 by expanding (x+h)^3, combining like terms, and canceling h to obtain 3x^2 + 3xh + h^2.

Compute the simplified difference quotient for f(x)=1/x by substituting x+h for x. By combining to a common denominator and canceling h, obtain -1/(x(x+h)).

Practice evaluating the difference quotient for f(x) = -2x^2 - 3x by substituting x+h, expanding, and simplifying, with careful attention to parentheses and canceling terms.

Learn that derivative is the slope of tangent line, defined as the limit of the difference quotient with f prime and dy/dx notation, noting differentiability failures like vertical tangents.

Apply the limit definition of the derivative to f(x)=x^2, simplify to f'(x)=2x. Evaluate f'(2)=4 and derive the tangent line at x=2 as y=4x-4.

Compute the derivative of f(x)=x^3 as f'(x)=3x^2, evaluate f'(1)=3 and f'(-2)=12, and derive the tangent line at x=1 using the point-slope form y-1=3(x-1), giving y=3x-2.

Use the definition of the derivative to compute f'(x) for f(x) = -2x - 4, show cancellations in the difference quotient, and obtain f'(x) = -2, with f'(3) = -2.

Compute the derivative of f(x) = -2x^2 - 3x via the limit, yielding f'(x) = -4x - 3 and f'(2) = -11, then the tangent line at x = 2 is y = -11x + 8.

Introduce the power rule for derivatives and clarify Leibniz notation, showing how to differentiate powers via the difference quotient and limits, including coefficients and negative exponents.

Apply the power rule to derivatives of negative exponents, shaping -3/x to 3/x^2. See how y = 1/x yields y' = -1/x^2 and how negative exponents move to the denominator.

Rewrite square and cube roots as exponents, apply power rule, and differentiate to get 1 over 2 square root of x and 1 over 3 cube root of x squared.

Derivatives of constants are zero, including square root of 5, then apply the sum and difference rules to differentiate terms like 3x^5, 2 cube root of x, and 1/(3x^2).

Differentiate fractional power terms using the sum and difference rule, apply exponent rules for x^(2/3), x^(1/2), and x^(-1/5), and rewrite with positive exponents.

Differentiate y = x^2 + 2x + 4 to get y' = 2x + 2; with slope 2 at x = 0, tangent line is y = 2x + 4.

Apply the product rule to derivatives of f(x) and g(x) when multiplied, taking the derivative of the first times the second plus the first times the derivative of the second.

Apply the product rule to differentiate f(x)=(3x-4)(x+6). Then f'(x)=3(x+6)+(3x-4)=6x+14, with optional factoring to 2(3x+7) for simplification.

Apply the product rule to differentiate a product, using f(x) and g(x), combining derivatives and original functions to obtain the final expression.

Apply the quotient rule to differentiate a function over another function, using f(x) over g(x). Compute the derivative as (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2.

Differentiate (3x^2+5)/(2x-1) using the quotient rule with f(x)=3x^2+5 and g(x)=2x-1. Compute (f'(x)g(x) - f(x)g'(x)) / g(x)^2 and discuss careful distribution and potential factoring for simplification.

Learn to differentiate a quotient by either using the quotient rule or simplifying it into separate fractions, then apply power rules for the derivative.

Master the chain rule for derivatives of function compositions by differentiating f(g(x)) with f(x)=x^3 and g(x)=x^2+2, using f'(g(x)) g'(x) and the derivative of the inside.

Apply the chain rule to differentiate (7x−16)^5 by viewing it as a composition f(g(x)) with f(x)=x^5 and g(x)=7x−16. Compute the derivative as f′(g(x))·g′(x)=5(7x−16)^4·7, yielding 35(7x−16)^4.

Apply the chain rule to differentiate the square root of 2x^2 minus 3 as (2x^2-3)^(1/2). The derivative equals 2x / sqrt(2x^2-3).

Apply the product rule and chain rule to differentiate f(x)=(x-3)^3(2x+1)^4, combine derivatives, and simplify the result via factoring.

Differentiate the expression ((2x-7)/(4x+3))^3 using chain rule on the outer power and quotient rule on the inner fraction, then simplify the result.

Explore higher order derivatives and their notations—f', f'', f''', f^n—and dy/dx, then practice computing multiple derivatives and their applications.

Apply successive derivatives to y = 3x^4 - 5x^2 with the power rule, and obtain the third derivative 72x as constants drop out.

Compute the third derivative of y = 1/x, using x^-1, and show how negative signs affect each step from first to third derivative, avoiding common mistakes.

Find the second derivative of (x^2 - 3)^6 by applying the chain rule to get 12x(x^2 - 3)^5, then use the product rule and simplify to 12(x^2 - 3)^4(11x^2 - 3).

Explore position, velocity, and acceleration through derivatives, including the second derivative of position. Apply ideas to solve problems and understand units like meters per second and meters per second squared.

Model kaiju movie sales with a cubic function; 2023 sales equal 262.6 million. N'(10) equals 70.21 million per year and N''(10) equals 13, showing increasing sales with acceleration.

Learn to graph functions from an equation by using derivatives to identify increasing and decreasing intervals, classify local and absolute extrema, and apply the first derivative test.

Apply the first derivative test to the cubic 2x^3-3x^2-12x+10, locate critical values -1 and 2, and identify a relative maximum at -1 and a relative minimum at 2.

Find the derivative of f(x)=2x^2-3x+5, locate the critical value x=3/4, compute y=31/8, and identify a relative minimum with increasing/decreasing intervals.

Explore relative extrema using the first derivative test; identify x = 0 as a critical value where the derivative does not exist, and conclude there are no relative extrema.

Explore the second derivative to determine concavity, identify points of inflection, and use sign changes to classify concave up or down on a graph.

Apply the second derivative test to classify extrema and identify concave up or down intervals, using critical values and points of inflection such as x = 1/2.

Apply the second derivative test to 2x^2 - 3x + 5 to show f''(x) = 4, confirm concavity up everywhere, and note there are no points of inflection.

Take the derivative to locate the critical value at x = 1; use the second derivative to identify a point of inflection at x = 1 and determine concavity.

Explore vertical and horizontal asymptotes and how graphs behave around them, including how denominator zeros create vertical asymptotes and how crossing a horizontal asymptote requires returning to it.

Explore horizontal asymptotes and oblique asymptotes by analyzing polynomial ratios, identify when y equals a over b, y equals zero, or a slant line, and apply degree comparisons.

Learn to find the slant asymptote of (x^2-9)/(x+2) using long division, yielding y = x − 2 and identifying the vertical asymptote at x = −2.

Identify x and y intercepts by setting y to zero and x to zero for the function (x^2-9)/(x+2), yielding x-intercepts at 3 and -3 and a y-intercept at (0,-9/2).

Sketches f(x) = -2/(x-5): no x-intercept, y-intercept 2/5, vertical asymptote at x=5, horizontal at y=0; left side concave up and increasing, right side concave down and increasing; no inflection points.

Analyze and sketch the function (2x+1)/x using intercepts, asymptotes, and derivatives to determine domain, increasing/decreasing behavior, and concavity, producing a complete graph.

Learn how absolute minimum and absolute maximum occur on closed intervals and why the extreme value theorem guarantees them.

Find the absolute extrema of f on [-1,2] by evaluating endpoints and critical points x=±2/3; the min is (2/3, 3.2) and the max is (2, 21).

Find the absolute maximum at (1,1) for f(x)=2x−x^2 on (−∞,∞) via the first and second derivatives; on the closed interval [−1,2], endpoints give the absolute min (−1,−3) and max (1,1).

Explore exponential and natural log functions with financial applications, from the future value formula A = P(1 + r/n)^{nt} to continuous compounding and the constant e.

Calculate the future value of a $2,500 investment at 0.0425 over seven years with annual, monthly, and daily compounding, using the formula p(1 + r/n)^(nt).

Demonstrates continuous compounding using the formula p e^{rt} applied to a 2500 principal at 0.0425 over seven years.

Explore the natural logarithm with base e, its inverse relationship to e, and key properties such as ln(e^k)=k, ln(1)=0, and ln(mn)=ln m+ln n, through examples like ln(e^4) and ln(e^-1).

Learn to solve exponential and logarithmic equations for x using natural log, illustrated by e^(4x)=5 and by dividing by 3 to handle the constant before applying ln.

Learn how present value uses continuous compounding to determine today’s investment needed to reach a goal, illustrated by needing about $29,577 today to have $50,000 in 15 years at 3.5%.

Explore the derivative rules for exponential functions with base e, including e^x, e^{2x}, and expressions like 3e^x. Learn how constants factor out and how cases like e^{2x} and e^{-x} differ.

Apply the constant multiple rule to derivatives of exponential functions: the derivative of 4e^x is 4e^x, same for 3e^x and -3e^x, with e^(2x) treated separately.

Apply the product rule to differentiate x^3 e^x, obtain 3x^2 e^x + x^3 e^x, and factor to x^2 e^x (3 + x) for practice.

Apply the quotient rule to differentiate e^x over x^2, factor out e^x and x, and simplify to e^x(x-2)/x^3.

Apply the chain rule to derivatives of exponential functions with general exponents, such as e^(f(x)), by multiplying e^(f(x)) by f'(x) and illustrating with e^(x^2).

Apply the chain rule to differentiate e to the negative 3x, using the derivative of -3x as -3, and obtain -3 e^{-3x}.

Differentiate e^(x^3 - x) using the chain rule with f(x) = x^3 - x. The derivative is (3x^2 - 1) e^(x^3 - x).

Differentiate e^{√x−2} using the chain rule, multiply by the derivative of the inner function, and rewrite the result as e^{√x−2}/(2√(x−2)).

Explore the derivative of the natural logarithm, including cases like 2 ln x and ln(2x), and learn d/dx[ln f(x)] = f'(x)/f(x) with constant placement and chain rule.

Differentiate a constant times the natural log of x, showing that the derivative equals six over x.

Apply the product rule to differentiate x^4 ln x, use the derivative of ln x as 1/x, and simplify to x^3(4 ln x + 1).

Apply the chain rule to differentiate the natural log of 5x, showing how 1/(5x) times 5 simplifies to 1/x.

Apply the chain rule to differentiate the natural log of x^3 minus 3, yielding 3x^2/(x^3-3). Avoid canceling x^2 with x^3-3, since they are not common factors.

Apply logarithm properties to differentiate the natural log of a quotient by rewriting ln((x^3-3)/(2x)) as ln(x^3-3) minus ln(2x), then use the rule d/dx ln f(x)=f'(x)/f(x) to get 3x^2/(x^3-3) minus 1/x.

Learn how revenue, costs, and profits relate in business calculus, form a profit function by subtracting costs from revenue, and compute marginal revenue and marginal profit as derivatives.

Derive the marginal profit for the profit function 3X - 0.5X ln X using the product rule and derivative rules, yielding P'(X) = 2.5 - 0.5 ln X.

Explore antiderivatives and integrated processes, the backward step of derivatives, and how the integral symbol, dx notation, and the power rule reveal the area under curves.

Apply the power rule to integrate x^3 and sqrt(x), yielding x^4/4 and 2x^(3/2)/3 plus C. The example reinforces working with fractional exponents and the role of constants in integration.

Apply the power rule to integrate 4x^5 dx, choosing to factor out the constant or keep it inside the integral, then simplify to (2/3)x^6 + C.

Apply the power rule to integrate 1/x^6 by converting to x^-6, add 1 to the exponent, divide by -5, and express the result as -1/(5x^5) + C.

Learn why the integral of 1 over x equals the natural log of the absolute value of x plus C, and when x is positive the absolute value isn’t needed.

Compute the integral of e to the x dx, with antiderivative e to the x plus c. Extending to e to the 2x or e to the 3x increases complexity.

Utilize the exponential rule to integrate e^(ax) dx, yielding (1/a) e^(ax) + c, and validate by differentiation; recognize the alternative u-substitution.

Explore integration by splitting the expression 5x^7 − 2x + 3 into three integrals and applying the power rule, handling constants, and combining results into a final antiderivative.

Integrate velocity with the initial height of 20 ft to obtain h(t) = -16t^2 + 60t + 20 and v(t) = -32t + 60; at t=3, h=56 ft, v=-36 ft/s.

Learn how definite integrals compute the area under a curve between two endpoints by evaluating F(b) minus F(a) and applying to examples like ∫_2^4 x^2 dx.

Calculate the definite integral of x squared from 0 to 2 by integrating to x to the third over 3 and evaluating at the bounds, yielding an area of 8/3.

this lecture explains how to evaluate a definite integral of x/4 from 1 to 5, showing factoring out 1/4 or handling the denominator directly, yielding a final value of 3.

Learn how to evaluate a definite integral from 2 to 4 of 3x^3 minus 5x minus 9 by combining terms, applying antiderivatives, and careful subtraction to reach 132.

Compute the total cost by evaluating the definite integral from 0 to 200 of the marginal cost function, -0.05x + 10, to obtain the cost for 200 pounds.

Evaluate the integral of e^{-x} from -2 to 3 using the 1/a rule with a = -1. Emphasize parentheses when plugging limits; form is e^3 + e^2, approx 7.34.

Evaluate the definite integral of 4 minus x squared from -1 to 4 by applying the antiderivative 4x minus x^3/3 and carefully evaluating at the bounds, yielding -5/3.

Explore substitution as a core integration technique, used when the derivative follows a chain rule, with varied problem types and practical use in business calculus and beyond.

Apply u-substitution to the integral of 2x^2(x^3-5)^6 dx by setting u = x^3-5, with du = 3x^2 dx, yielding (2/21)(x^3-5)^7 + C.

Set u equal to x squared plus seven with du equal to 2x dx. Rewrite integral as u to the tenth, obtaining u to the eleventh over eleven, then back-substitute.

Apply u-substitution in business calculus to the integral x^5 e^{x^6} dx, with u = x^6, yielding du = 6x^5 dx and reducing to (1/6) e^{x^6} + c.

Set u = 4 + 2x, so du = 2 dx. Then integral becomes 1/2 ln(u) + c, and substitute back to 1/2 ln(4+2x) + c.