Calculus 1

learn to evaluate limits as x approaches zero by converting to simplest form, handling 0/0 indeterminate cases, and using a substitution to express the limit as ln(1+y)/y, yielding 1.

Examine the limit of sin x over x as x approaches zero, using inequalities and the squeeze theorem, to show the limit equals one for both positive and negative x.

This lecture solves a limit as x approaches infinity by substituting y = x - 1 and using natural logs to analyze ln(1+y)/y as y grows.

Compute the limit of (1 + 1/x)^x as x approaches infinity, illustrating the standard limit concept in calculus.

Compute the limit of a rational function as x approaches plus or minus infinity by canceling factors and comparing leading terms; the limit evaluates to 2.

explains evaluating a limit with absolute value as x approaches 3 from left and right, showing sign changes and that the left-hand limit diverges to negative infinity.

Prove the continuity at zero of the piecewise function f(x)=x for x≠0 and f(0)=0 using the epsilon-delta definition, showing |f(x)-f(0)|<epsilon whenever |x|<delta.

This lecture investigates the continuity of f(x) = x sin(1/x) for x not equal to 0 with f(0) = 0, applying the epsilon-delta definition to establish continuity.

Calculus 1 continuity example 13 finds c so that f is continuous on [0,1) by setting f(1)=c equal to the limit of (1−x^2)/(x−1) as x→1, which equals −1/2.

examine a piecewise function with f(x)=x+2 for x<1, f(x)=x for 1≤x<2, and f(x)=x+5 for x≥2, and identify discontinuities at x=1 and x=2 by comparing left and right limits.

Analyze continuity at x = 1 by equating the left and right limits, which yields a + b = 3 since the function is continuous.

Examine the continuity at zero for a piecewise function: F(x)=1+x for x≠0 and F(0)=1, where the limit as x→0 is 1, matching F(0).

Analyze continuity at zero by examining left and right limits as x approaches zero of two piecewise functions, showing that the function value differs from its limit, hence discontinuity.

Learn to differentiate an equation involving x and y with natural logarithm and sine, applying product and chain rules to find dy/dx.

Differentiate y = f(v) with respect to x by applying the quotient rule to (x^2-1) and the chain rule, giving dy/dx = f'(v) * dv/dx.

Differentiate y = 1 − x with respect to x to obtain -1, and explore the derivative forms involving (1 − x) squared.

Explore derivatives in Calculus 1 by examining inverse sine and natural logarithm expressions, applying differentiation with respect to x and identifying key results such as 1/(1-x^2).

Differentiate the equation xy - (x + y) = 0 with respect to x, solve for dy/dx, and simplify using terms involving sine(x+y).

Practice differentiating the function x^2 plus y with respect to x, using dy/dx to relate x and y and obtain the derivative expression.

Differentiate the implicit relation x^2 + y^2 = 1 with respect to x using the product and chain rules, then solve for dy/dx.

Combine trigonometric and inverse trigonometric terms to explore derivatives, simplify expressions like one minus x squared and one plus y squared, and derive y prime.

Explore nth derivatives by computing the first, second, and third derivatives of y = e^x, and observe the consistent exponential pattern across derivatives.

Explore nth derivatives and the Leibniz rule through differentiating products, applying Leibniz theorem, and using binomial coefficients and factorial properties to simplify expressions.

Apply the Leibniz rule to y equals sign inverse x^2, differentiate to relate y' and y'', and derive a differential equation with (1 - x^2) and y'.

Explore nth derivatives using a special formula related to Leibniz rule, transforming expressions into compatible forms like 1/(x±a) and x±a, and apply the rule to differentiate step by step.

Master nth derivatives using the Leibniz rule for products in Calculus 1, exploring derivatives of u and v and factorial patterns across examples.

This lecture demonstrates differentiating f(x)=ln(1+x^2) and applying Leibniz rule to verify a derivative identity that equals zero, involving f', f'', and higher derivatives.

This lecture presents an exercise on nth derivatives and Leibniz rule, showing how to differentiate expressions involving x and y and their derivatives.

Illustrate functions of several variables by defining z = f(x, y) for two variables and z = f(x, y, z) for three, using rectangle area and rectangular prism volume.

evaluate the limit of a two-variable function as (x, y) approaches (0, 0), using numerator 5 − x^2 and denominator 4 + x + y, yielding 5/4.

evaluate the limit of a multivariable rational function as (x,y) approaches (2,2), simplify the numerator and denominator, and obtain 1/6.

The lecture shows the limit at (0,0) does not exist for the function, because line analysis yields a m-dependent limit and polar-coordinates analysis yields a theta-dependent result.

Show how limits of functions of several variables approach (0,0), reveal path dependence along lines y = a x, and demonstrate that the limit may not exist.

Compute partial derivatives of a two-variable function in calculus 1, finding f(x,y) with respect to x and with respect to y, including x^2/(x^2+y^2) and 1/x.

Compute partial derivatives of f(x,y) = e^x sin(By) with respect to x and y, treating the other variable as constant, yielding ∂f/∂x = e^x sin(By) and ∂f/∂y = B e^x cos(By).

Explore partial derivatives of a function f(x,y) involving the natural log of x^2 + y^2, computing ∂f/∂x and ∂f/∂y and simplifying to reveal cancellations.

Compute partial derivatives of V = x y; Vx = y, Vy = x, and derivatives Vxx = 0, Vyy = 0, with mixed derivatives Vxy = Vyx = 1.

Compute the first partial derivatives with respect to x and y for a function involving 1 minus x y, then determine the second partial derivatives.

This exercise walks through computing the partial derivatives fx and fy of f(x,y) and demonstrates that f_xy equals f_yx, using terms like x y, cos(bx + c), and sin(bx + c).

Compute the partial derivatives of F(x,y) with respect to x and y, treating the other variable as constant, and determine F_x and F_y for the given function.

This lecture explores computing partial derivatives of a function f(x,y) with respect to x and with respect to y, using sine and cosine relationships and related derivative rules.

Compute partial derivatives fx and fy for f(x,y) = (x^2 + y^2)/(xy), then compare mixed derivatives f_xy and f_yx, showing equality.

Practice partial derivatives through exercise 33 in calculus 1, reinforcing problem-solving skills and familiarity with the mechanics of partial derivatives.

Investigate partial derivatives and how signs and constants affect differentiation, using clear examples to reinforce when a variable remains constant and how sign changes influence results.

Walks through mean value theorem exercises, checks endpoint values and derivative conditions, and finds c in each interval, illustrating how f(b) - f(a) = f'(c)(b - a).

Learn how to resolve 0/0 indeterminate forms with L'Hôpital's rule by using derivatives of two functions f and g to evaluate limits as x approaches a.

Master evaluating limits that form 0/0 using l'Hôpital's rule, performing successive derivatives and algebraic simplifications. Tackle exercises on limits with trigonometric and hyperbolic expressions, ensuring accurate cancellation and convergence.

Practice evaluating 0/0 and infinity limits with l'hôpital's rule by deriving log, sine, and cosine functions to resolve the limits in exercise 4.

Apply L'Hôpital's rule to 0/0 indeterminate limits, differentiate the numerator and denominator, and compute limits as x approaches zero using sin x, cos x, and related expressions.

This lecture demonstrates solving 0/0 limits with l'Hôpital's rule in calculus 1, showing alpha = -2 yields a finite limit of -1, and discusses natural-log cases.

This lecture demonstrates evaluating indeterminate limits using l'Hôpital's rule, converting 0/0 and infinity/infinity forms into derivatives, with examples involving natural logarithm, sine, cosine, and limits as x approaches 0.

Explore evaluating limits using L'Hôpital's rule for 0/0 forms and other indeterminate cases, with derivatives, logarithmic and trigonometric expressions.

Demonstrate solving 0/0 and infinity-form limits with L'Hôpital's rule and rationalization. Apply to limits as x approaches infinity and 1, yielding results like 3 and -1.

This lecture demonstrates solving 0/0 indeterminate forms with l'Hôpital's rule, showing two limits: one with hyperbolic functions and another using natural log to derive a result.

Explore integration formulas for powers, exponential and logarithmic functions, and trigonometric and hyperbolic functions, including antiderivatives of sine, cosine, secant, cosecant, and related identities.

Explore integration by formula, convert improper integrals to proper forms, and apply substitutions to evaluate antiderivatives, including logarithmic cases.

Examine the cone formed by lines from a circle’s circumference meeting at a fixed point, its axis, and how a plane cuts the cone to produce ellipse, parabola, and hyperbola.

examine parabola problems by converting to standard form, locate the vertex and focus, and use axes and vectors to draw and analyze the parabola.

Explore a hyperbola exercise by locating the center at the origin, deriving the asymptotes, and identifying the foci. Then graph the hyperbola.