Calculus 1, part 2 of 2: Derivatives with applications

Build a foundation in calculus by learning derivative terminology and notation. Study delta, differential, tangent lines, and multiple derivative notations such as prime and dy/dx.

Reinforce how straight lines and their slopes underlie derivatives, and point you to precalculus resources on slope, average and instantaneous rate of change, and OpenStax references.

determine the points where derivatives exist by ensuring x0 lies in the domain and is an accumulation point, and distinguish interior, left, and right derivatives at domain endpoints.

Define the derivative at a point as the limit of the difference quotient, giving the instantaneous rate of change and the tangent slope; differentiability requires this limit to exist.

Learn two methods to find tangent line equations using the derivative, including point-slope form and slope-intercept form, with f'(x0) as the tangent slope.

Derive derivatives from the definition for linear functions, showing constant functions have derivative zero and lines with slope m have derivative m. Link the derivative to the slope of y=mx+b.

Compute the derivative of f(x)=x^2 at x0=1 via the difference quotient, find f'(1)=2, and determine the tangent line y=2x-1 through (1,1).

Compute the derivative at x0 = 4 for the quadratic f(x) = -x^2 + 4x - 3 and determine its tangent line, giving y = -4x + 13.

Compute the derivative of f(x) = 1/3 x^3 at x0 = -1 using the difference of cubes, then state the tangent line y = x + 2/3.

Examine the derivative of the square root function at zero with the right-hand quotient, showing nonexistence due to a vertical tangent, and why a zero derivative invalidates the inverse's derivative.

Compare two equivalent derivative definitions via difference quotients, using h instead of delta x, and illustrate with f(x)=x^2 at x0=1.

This lecture examines the absolute value function, continuous everywhere but not differentiable at zero, illustrating a cusp and the left and right derivatives that fail to match.

Explore how absolute values generalize differentiability for a differentiable function, revealing when the absolute value preserves or reverses the derivative, and when zero values create cusps.

Present an iff characterization of differentiability at a point: f(x) - f(x0) = a(x - x0) + r(x,x0) with r(x,x0)/(x - x0) → 0, enabling linearization along the tangent line.

Show that differentiability implies continuity, and that continuity is a necessary condition, not sufficient, for differentiability. Illustrate with the floor and absolute value examples that continuity does not guarantee differentiability.

Show that for differentiable f with f(x0) ≠ 0, |f| is differentiable at x0; its derivative is ± f'(x0) depending on the sign of f(x0), via continuity and separation lemma.

Explore a proof of part c2.1 from video 21, showing that if f'(x0)=0 then (|f|)'(x0)=0, using the squeeze theorem and case analysis.

Analyze the piecewise function f(x)=x|x|, show differentiability at zero with f'(0)=0, and derive the derivative as 2x for x>0 and -2x for x<0.

Analyze the differentiability of a piecewise function by comparing left and right derivatives across intervals and at the meeting points -1 and 1.

Recognize derivatives through a limit as a difference quotient for f(x)=1/(x+1) at x0=1, finding f'(1)=-1/4 and the tangent line y=-1/4 x+3/4 with the graph and its vertical asymptote at x=-1.

Recognising derivatives and applying a conjugate trick, the lecture derives f'(0)=1/2 for f(x)=sqrt(1+x) and finds the tangent line y=1/2 x+1 at x0=0.

Recognize derivatives using limits: apply sin x over x and (e^x−1)/x to find derivatives at zero, illustrate derivatives of e^x and sin x with their tangent lines.

Compute derivatives from the definition for a function involving arcsin and a square root, demonstrating continuity and limit cancellation at x = 1, and conclude f′(1) = 1 + π/4.

Show that the tangent to f(x)=1/x at (a,1/a) forms a right triangle with the axes; compute its slope -1/a^2, intercepts 2a and 2/a, and area equals 2.

Define higher order derivatives and their notation, compare primes, parentheses, Leibniz, and D, and note when derivatives exist, along with examples like C∞ functions and Taylor polynomials.

Explore the geometric interpretation of differentials, linking infinitesimal changes to tangent-line linearization and the derivative, and prepare for practical linearization in later videos.

Learn how linearization uses the tangent line at a point to approximate a differentiable function near that point, with errors shrinking as h approaches zero.

Explore how linearization around x0 = 4 yields a local tangent-line approximation for sqrt x, used to estimate sqrt(4.01), and why it fails for 9, 16, and 25.

Outline the plan for section three, the course's largest part, detailing how we derive derivatives of elementary functions using limits and rules, and noting 45 videos and future topics.

Derive the derivative of monic monomials x^n using power function type one, method one, and apply the difference of nth powers to obtain f'(x) = n x^{n-1}.

This lecture presents method two for the derivative of monic monomials (power functions type one) using the binomial theorem to prove that f'(x) = n x^{n-1} via a limit.

Compute the derivative of f(x) = x^{1/n} for n ≥ 2, noting domain restrictions: x>0 if n even, and x ≠ 0 otherwise. The derivative is (1/n) x^{1/n−1}.

Explain the derivative of power functions with negative integer exponents using method 1 and a reciprocal rewrite, and state the power rule f'(x)=k x^{k-1} for non-negative k.

Demonstrate the derivative of sine using method 1, the sine difference formula, and the standard limit sin x / x → 1, establishing f'(x) = cos x and tangent slopes.

Derive the derivative of cosine using the cosine difference formula (method 1) and the standard limit at zero, showing d/dx cos x = - sin x and its tangent slopes.

Explore the derivative of sine and cosine using method two, employing trigonometric identities and standard limits to derive the same results.

Apply substitution x = sin t to derive that the derivative of arc sine x equals 1/√(1−x²) for x in (-1,1), and note non-differentiability at ±1.

Learn the derivative of arc cosine, showing that d/dx arc cosine x equals minus one over the square root of one minus x squared for x in (-1,1), excluding endpoints.

Demonstrate the derivative of the exponential function e^x and show it equals itself using the limit. Derive f'(x)=e^x and the tangent line at zero, y=x+1.

Derive the derivative of the natural logarithm using limit definitions and logarithm rules, establishing d/dx ln x = 1/x for x>0 and graphically motivating d/dx ln|x| = 1/x.

Derive the derivative of logarithms with any base using a limit and base switch, yielding f'(x) = 1/(x ln a) for x > 0.

Master the rules of differentiation, the main theorem, and learn to differentiate almost any elementary function using the sum, product, quotient, and reciprocal rules, with sine, cosine, and e^x examples.

Explore the sum rule for derivatives through visual illustrations and a practical example, showing that the derivative of f+g equals f'+g' at a point.

Explore two visual illustrations of the product rule for derivatives, defining f(x)g(x) and deriving f'(x)g(x)+f(x)g'(x) through geometric rectangle pictures and precalculus examples.

Explore three proofs of the sum rule for derivatives, using difference quotients, h tends to zero, and delta notation, showing that f and g differentiable at x0 imply (f+g)'(x0)=f'(x0)+g'(x0).

Demonstrate the scaling rule for derivatives through three proofs, showing that the derivative of c f equals c times f', with examples and limit, h, and delta approaches.

Demonstrate the linearity of the differential operator: the derivative distributes over sums and scalar multiples. Use induction to extend this to any finite number of differentiable functions.

Show the product rule for differentiable functions by a limit of the difference quotient for f g, using the adding and subtracting trick to obtain f' g plus f g'.

Demonstrate that the derivative of x^n is n x^{n-1} for positive integers n using the product rule. Show that a polynomial's derivative is a polynomial of one less degree.

Proves the quotient rule and the reciprocal rule for differentiable f and g with g(x0) nonzero, using the derivative definition and limit laws.

Derive the power function derivative for negative integer exponents using the reciprocal rule, showing f(x)=x^k yields f'(x)=k x^{k-1} through multiple approaches.

Apply the quotient rule to show derivatives of rational functions are rational, as in f(x)= x/(x^2+1); factoring the result aids graphing and sign analysis while noting the domain excludes d(x)=0.

Generalize the product rule from two functions to a product of n differentiable functions, using the derivative operator and induction, with examples for four functions.

Apply the quotient rule to tan x = sin x / cos x using sine and cosine derivatives. Obtain d/dx tan x as 1/cos^2 x or 1 + tan^2 x.

Derive the derivative of arctangent, showing that d/dx arctangent x = 1/(1+x^2) for all real x, using a tangent substitution and the inverse relation.

Practice differentiating a sum of terms using linearity, the sum and scaling rules, and derivatives of basic elementary functions in Exercise 1.

Differentiate a sum of power functions by applying the sum rule and power rule, rewriting terms as x^alpha (x, x^{1/2}, x^{1/3}), and identify domain and differentiability.

Apply the sum rule and scaling rule to differentiate a sum of three power terms, derive and simplify, revealing a derivative negative for all x ≠ 0.

Practice differentiating with quotient and product rules, using f(2)=2 and f'(2)=3 to evaluate derivatives of x^2/f(x) and x^2 f(x) at x=2.

Two methods differentiate a quotient of polynomials: the quotient rule and splitting the fraction into power functions, with both methods yielding the same result.

Apply the product rule to differentiate a product of functions and compute y'(9) for y = u v using u(9)=2, u'(9)=-5, v(9)=1, v'(9)=3, yielding 1.

Practice differentiation of products using the product rule on x^2 sin x and f times arctan x, applying the two-function rule, then the three-function extension, with arctangent derivative 1/(1+x^2).

Practice differentiating a quotient of two functions with the quotient rule on a rational function, then simplify by factoring and cancellations to obtain a sign-analyzable, concise derivative.

Practice exercise nine differentiates a polynomial given as a product of two polynomials using the product rule. It also shows an alternate approach by expanding the product and applying the sum and scaling rules, yielding the same derivative 5x^4 + 3x^2 + 8x.

Compare differentiation by definition and by the quotient rule for a function, showing how the quotient rule yields the derivative (1−x^2)/(1+x^2)^2.

Analyze the differentiability of a piecewise function, exercise 11, showing a cusp at x = 2 with left and right derivatives -3 and -4, hence not differentiable there.

Derive the derivative of sin(3x) from the definition, showing it equals 3 cos(3x) rather than cos(3x), and highlight the role of the standard limit at zero.

Assess the differentiability of a piecewise function in exercise 12, locating cusps at x = 0 and x = pi/2 and applying the quotient rule and sine derivative.

Explore how polynomials with multiple zeros shape graphs: odd multiplicities cross, even touch the axis, and the derivative has a zero at x0 with multiplicity k-1.

Explore a non-negative fourth-degree polynomial with zeros at ±1 and p(0)=1, show that p(x) = (x-1)^2(x+1)^2, and compute p(2)=9 using two approaches (derivative-based and computational).

Explore differentiability and derivative graphs of four piecewise and polynomial functions, compute their derivatives, and sketch tangent slopes to connect geometry with formulas.

Use the quotient rule to find tangent lines to the rational curve y=(x-1)/(x+1) through (-1,0); the unique tangent has slope 1/8 and equation y=1/8 x+1/8 at x=3.

Find where to practice derivatives with eight solved problems and article solutions, and learn to construct functions to select the appropriate differentiation rules, including the upcoming chain rule for composites.

Explore section four on the chain rule, its use for derivatives of composite functions, and recommended reading that links derivatives to velocity and acceleration.

Reinforce the repetition of the composition of functions, with inner and outer functions, f applied first and g second, and domain requirements as a precalculus refresher.

Explore the derivative of composite functions and graph transformations involving scaling the argument, using inner and outer functions and the chain rule to explain sine and cosine behavior.

Explore the chain rule: state the theorem for f and g, demonstrate the derivative of a composition, and review the proof with examples like cosine of 5x and arctangent.

Learn how to generalize the chain rule to compositions of any number of differentiable functions, with notation, rationale, and an example using a polynomial, cosine, and square root.

Identify composite functions and apply the chain rule to differentiate examples, including polynomials, trigonometric functions, sums, products, and quotients, while examining domains and nondifferentiability at zero.

Examine how the order of functions in a composition changes the function and its derivative, applying the chain rule to two or three functions with examples like sine and logarithm.

Derive the derivative of x^k for negative integers using the chain rule, showing how to obtain f'(x)=k x^{k-1} with x ≠ 0.

Apply the chain rule to power functions with rational exponents to derive d/dx x^q = q x^{q-1}, extending the power rule from integers to all rationals.

Explore precalculus four formulas used in calculus one part two, including product and quotient rules for powers, the power rule, and the inverse relationship between exponentials and logarithms for differentiation.

Differentiate power functions with any real exponent, including irrationals, using two methods shown in the lecture. Apply the chain rule and exponential/logarithm substitution to obtain f'(x) = alpha x^{alpha-1}.

The lecture derives the derivative of exponential functions with base a>0, a≠1, showing d/dx a^x = ln(a) a^x by rewriting a^x as e^{x ln a} and using the chain rule.

Differentiates functions with both variable base and exponent, f(x)^{g(x)}, when f is positive, by rewriting as e^{g(x) ln f(x)} and applying the chain and product rules, with x^{1/x} as an example.

Explore the chain rule in calculus 1, derivatives with applications, through two classic examples: the derivative of log|x| equals 1/x for x≠0, and the derivative of (x−x0)^k equals k(x−x0)^{k−1}.

Learn how the chain rule proves differentiability of a composition and compute the derivative at zero using the given values f(0)=1, f'(0)=2, and g'(1)=3.

Master differentiating easy and complex functions by applying chain, product, and quotient rules, analyzing function construction with diagrams and extensive practice problems.

Practice differentiation by applying chain rule and product rule to a product of polynomial functions. Analyze derivative signs to determine where the function increases, decreases, and local extrema.

Differentiate a product of a polynomial and a square root using the product and chain rules, simplify to (2x^2+1)/sqrt(1+x^2), and note the derivative is always positive.

Apply the chain rule and quotient rule to differentiate a composition with a cubic root, derive a nonnegative derivative, and identify domain restrictions excluding one and minus one.

Practice differentiating a product of two composite functions using the product and chain rules, involving cosine squared and sine components, then simplify with double-angle identities.

Differentiates the composition f(x)=arcsin(sin x) using the chain rule, showing f'(x)=cos x/|cos x|=sign(cos x), undefined at x=pi/2+k pi, with derivative ±1 where defined.

Practice in differentiation of a tangent-based function, applying Pythagorean identity, sum and chain rules, and power rules to obtain derivative 1+tan^6 x on cos x ≠ 0.

Practice in differentiation exercise seven applies the chain rule and the power rule to a function with multiple square roots, derives the derivatives, and notes domain and positivity.

Revisit a precalculus construction of a multi-step function and practice differentiating it using the chain rule, sum rule, and power derivatives, illustrating domain considerations.

Differentiate the given rational function with the quotient rule and chain rule, applying arctan's derivative 1/(1+x^2). Note the domain gap at x=1 and that this exercise will be revisited later.

Practice differentiating a composite function with arctan and arcsin, applying chain and quotient rules; examine domains and show f′(x)=0 for |x|≥1, so the function is constant on (-∞,-1) and (1,∞).

Derive the derivatives of hyperbolic cosine, hyperbolic sine, and hyperbolic tangent from exponential definitions, using the quotient rule and identities, highlighting parallels to trigonometric derivatives.

Compute the derivative of the inverse hyperbolic sine from exercise 12 and compare it to arc sine, noting there is no minus sign and the result is 1 over sqrt(x^2+1).

Explore related rates and the chain rule, linking the rate of change of volume to radius in time via a composite function, with ball, balloon, and melting ice cube examples.

Use related rates on a spherical balloon: with dv/dt = 20 cm^3/s and r = 30 cm, find dr/dt = 1/(180 pi) cm/s using v = 4/3 pi r^3.

Treat the diameter as x in this related rates problem. Compute dV/dt using V=(pi/6)x^3, dV/dt=(pi/2)x^2 dx/dt; at x=6 cm and dx/dt=-0.5 cm/h, dV/dt=-9 pi cm^3/h.

Apply product rule to a related rates problem: compute rate of change of a rectangle's area when one side grows at 2 cm/s and the other shrinks at 3 cm/s.

Apply the chain rule to related rates on leaky inverted cone, relate radius to height with similar triangles, and use volume formula to find dh/dt at h = 4 m.

Solve a related rates problem for a water-filled cone, using similar triangles to relate volume and height, and find dh/dt = 1/(90 pi) m/min when h = 4 m.

Practice extensively with the article and book to build fluency in derivatives, using solved problems, diagrams, and a dedicated problem-solving article to verify your work.

Explore the intuition behind the derivative of inverse functions and differentiable functions, using geometric interpretation, symmetry about y=x, and the reciprocal slope relationship between f and f^{-1}.

Present the theorem on the derivative of the inverse function, outline its proofs and applications, and explain when the inverse is differentiable.

Compute the inverse of f(x) = 1 + 2/x on its domain, derive f^{-1}(x) = 2/(x-1), and verify the derivative of the inverse matches 1/f'(f^{-1}(x)) using the chain rule.

Use the inverse derivative formula for a strictly increasing differentiable f on r+, showing its inverse is differentiable and the derivative at two is 1/3 from f(1)=2 and f'(1)=3.

Derive the derivative of the nth root using the inverse of x^n, applying the derivative of inverse functions (method two) and power rules to handle even and odd n.

Explore method two for deriving the derivative of the natural logarithm, showing the inverse of the exponential function has derivative 1/x for x>0 using the theorem from video 118.

Develop an alternative derivation for the derivative of arcsin using the inverse function theorem and the positive cosine on (-pi/2, pi/2). Derive arcsin'(y) = 1/√(1−y^2) for y in (-1,1).

Method two confirms that the derivative of arctangent, as the inverse of tangent, equals 1/(1+x^2), using the inverse function theorem and f'(x) = 1 + tan^2 x.

Explore intercept equations of straight lines, convert from slope-intercept form, and analyze symmetry about the line y=x, including intersection points (a b)/(a+b) and the zero-sum case a+b=0.

This optional lecture shows that for differentiable invertible functions f and its inverse, the tangent lines at corresponding points are symmetric about y = x, with intercepts swapped.

Compute the derivative of the inverse for f(x)=x^3+x and confirm f is 1-to-1 on R using three methods. Find (f^{-1})'(10)=1/13 and the tangent lines at (2,10) and (10,2).

Apply the derivative of an inverse to a fifth-degree polynomial by proving invertibility, then compute g'(35) = 1/f'(2) = 1/81.

Analyze a cubic on [0,4], prove invertibility, identify f and f inverse domains and ranges, and compute (f^{-1})'(2) = -1/9 using f'(1) = -9.

Bridge theory and practice by formalizing derivative concepts with theorems and applications, covering monotonicity, critical points, optimization, related rates, and the role of the second derivative and convexity.

Explore how maximum and minimum concepts apply to general functions, not just continuous ones. Learn about global and local extrema, boundedness, and examples using arctan and tangent.

Explore how monotonicity extends beyond continuous and differentiable functions, showing why continuity and invertibility on an interval imply strict monotonicity, while invertible functions need not be monotone.

The lemma states that a positive derivative at an interior point makes f increase to the right and decrease to the left; a negative derivative reverses this, using epsilon-delta neighborhoods.

Show that for a differentiable f with f(x0)=0 and f'(x0) ≠ 0, |f| is not differentiable at x0, via previous video lemma, yielding cusps with opposite slopes m and -m.

Fermat's theorem provides a necessary condition: at interior extrema of a differentiable function, the derivative must be zero, revealing stationary points.

Identify critical (stationary) points, singular points, and plateaus when the derivative equals zero or is undefined, and relate them to local and global extrema and non-differentiable points.

Discover Rolle's theorem: a continuous function on [a, b] with equal endpoints has stationary point in (a, b). The proof uses min-max and Fermat's theorems, including constant and nonconstant cases.

Apply Rolle's theorem to a degree five polynomial with zeros at 1–5, proving its derivative has four zeros between zeros and generalizing to any differentiable function vanishing at n points.

Use Rolle's theorem on the polynomial g with equal endpoints to guarantee a c in (0,1) where g'(c)=0, and compare with a manual derivative approach using the product rule.

Explore the mean value theorem (Lagrange), proving that a secant slope equals a tangent slope via Rolle’s theorem and revealing its geometric interpretation and links to Taylor polynomials.

Apply lagrange's theorem to find c in (1,2) where f'(c) equals the slope of the secant through (1,1) and (2,1/2) for f(x)=1/x, yielding c = sqrt(2).

Apply Lagrange's theorem to prove sin x < x for all positive x, using the difference quotient and cosine bound.

apply Lagrange's theorem to prove sqrt(1+x) ≤ 1 + x/2 for x ≥ -1, with equality at x=0, in example 3; use two ranges and visualize the tangent bound.

Explore Cauchy’s extended mean value theorem, prove it via a Rolle-based construction, and see how Lagrange’s and Rolle’s theorems interrelate with applications to derivatives.

Examine the Darboux property for derivatives on a closed interval, showing that for any d between f'(a) and f'(b), there exists a c in (a,b) such that f'(c) = d.

Explore the Darboux property for derivatives through examples, including signum and floor functions, and learn to solve a simple differential equation by integrating terms and verifying solutions.

Determine monotonicity of differentiable functions by evaluating derivative signs on the interval. Show non-negative derivatives imply non-decreasing, non-positive imply non-increasing, with a short Lagrange-based proof and the 1/x example.

Use the derivative test to confirm monotonicity of differentiable functions, with positive derivatives signaling increasing and negative signaling decreasing, illustrated by e^x, log x, arctan, arcsin, and x^alpha.

Demonstrate the monotonicity of exponential functions using derivatives: bases greater than one are strictly increasing on all real numbers, while bases between zero and one are strictly decreasing.

Revisit derivatives of inverse functions, prove invertibility via monotonicity, and compute the derivative of the inverse at two over pi using the quotient rule and inverse differentiation.

Prove that derivative zero on an interval implies the function is constant there, and conversely. The proof uses Lagrange's theorem to link the derivative to the secant slope.

Note that two differentiable functions with the same derivative on an interval differ by a constant, so their antiderivatives form a constant-shifted family, the indefinite integral.

Explain how two functions can share derivatives on intervals yet differ by constants due to a domain gap at x = 1, arctan involved, so there is no contradiction.

Derivatives are zero outside (-1,1), so the function is constant there; on (-1,1) it equals 4 arctan x, with f(-1) = -pi and f(1) = pi.

Learn how to rewrite a function as a piecewise form using trigonometric identities, turning a derivative problem into a three-region piecewise expression with -pi, 4 arctan x, and pi.

Apply the first derivative to study monotonicity and optimization, locating critical points and intervals of increase or decrease, with polynomial examples using derivative factoring to find local maxima and minima.

Explore polynomials through precalculus revisited, covering factoring, zeros, and derivatives to locate critical points, assess monotonicity, and sketch graphs for various degree cases.

discover a quick derivative-based method to find the vertex of a parabola, with x-coordinate -b/(2a) and y-coordinate -delta/(4a). Link precalculus ideas and monotonicity with calculus.

Investigate a rational function by applying the quotient rule to locate derivative zeros, determine local and global extremes, and intervals of monotonicity, and review domain and asymptotes.

Determine local and global extrema and intervals of monotonicity for a two-square-root function using its derivative, with horizontal asymptotes at y equals one half and y equals minus one half.

Examine a composite arctan of a rational function to determine domain, horizontal asymptotes, and monotonicity; derive f'(x) and show strictly increasing with no local or global extrema.

Analyze why the derivative offers limited insight for a function with a restricted domain. Compute the quotient derivative of arc sine and tangent and note the global maximum.

The function has a one-sided vertical asymptote at x = -1, is decreasing on (-∞, -1) and (-1, ∞), with infimum 0 and supremum ∞, and no global extrema.

Apply the first derivative test to classify critical points by analyzing sign changes in the derivative, identifying local minima, maxima, and plateaus, and connect to Fermat's theorem and optimization.

Use the second derivative test for twice differentiable functions: negative implies a local max, positive a local min, and zero leaves it indecisive (contrast with the first derivative test).

Compare the first derivative test and the second derivative test for classifying critical points where the derivative equals zero, noting their advantages, disadvantages, and when they yield extrema.

Optimize a continuous function on compact or non-compact domains by locating critical and singular points, evaluating end points, and applying the min-max theorem to find global extrema.

Analyze an optimization on a closed interval for f(x)=| (x-2)(x+1) |, identifying end points, singular points, and critical points to reveal global and local extrema.

Compute the derivative of f(x)=8−9^x−2·3^x to show it is strictly decreasing. Conclude range is (-infinity, 8), with 8 not attained, and as x tends to infinity, f(x) tends to -infinity.

Explore optimization of the exponential–polynomial f(x)=49^x−7^x−6 using derivatives to determine its range, with a global minimum at x=log_7(1/2) and range [-6 1/4, ∞).

On a closed interval, use the min-max theorem and derivatives to find the function's range: endpoints give 8 and a small positive value, while the critical point x=2 yields 16.

Evaluate endpoints and the interior critical point of the function on a closed interval using the derivative. The maximum occurs at x = 1/2 and minimum at x = -1/2.

Explore how derivatives compare numbers by analyzing f(x)=x^(1/x) for x>0, showing a unique maximum at x=e and using this to compare e^pi and pi^e.

Apply calculus to compare numbers in problem 7 by analyzing f(x)=log_x(x+1) for x>1, rewrite with natural logs, differentiate to show f is strictly decreasing, proving log_{n-1}(n) > log_n(n+1) for n≥3.

Explore convexity and concavity for twice differentiable functions, using second derivatives to characterize shapes and identify inflection points, and see how tangent line behavior and secants illustrate these ideas.

analyze convexity and concavity across functions using second derivatives, identify inflection points and critical inflection points, and compare monotonicity with power, exponential, logarithmic, and trigonometric examples.

Explore convexity and concavity via the second derivative, identify inflection points at 2 minus sqrt(2) and 2 plus sqrt(2), and note that x^2/e^x tends to zero as x grows.

Analyze the second derivative to determine convexity and concavity, locating the inflection point at x = log base seven of one fourth; concave to the left, convex to the right.

Analyze convexity and inflection of the function, showing second derivative zeros at x = -1/2; convex to the left and concave to the right, with a domain break at -1.

Learn L'Hôpital's rule, its theorem with a detailed article proof, and how to use it for indeterminate limits. Practice exercises show form reductions and cautions about determinate cases.

Apply L'Hôpital's rule to zero-by-zero and infinity-by-infinity limits, using derivatives and neighborhood conditions, illustrated with x→1 and x→0 examples.

Apply l'Hôpital's rule to infinity over infinity in two limits, showing logarithm grows slowest, exponentials fastest; differentiate repeatedly until constants appear, yielding both limits equal to zero.

Explore how to solve an infinity times zero indeterminate form using two rewrites and L'Hôpital's rule, showing the limit equals zero.

Apply l'Hôpital's rule to an infinity minus infinity limit, verify the indeterminate form, rewrite as a single fraction with logarithm terms, and differentiate twice to obtain minus one half.

Apply l'Hôpital's rule to the 0/0 limit of x raised to sin x by rewriting as e to the sin x ln x, and conclude the limit equals 1.

Apply l'Hôpital's rule to the indeterminate form and compute the limit as x tends to zero from the right of e^{-sin x log x}, revealing the reciprocal relation with 1/(x^2 sin x).

Apply l'Hôpital's rule to evaluate the limit of x^(1/(1−x)) as x approaches 1. Rewrite as e^{(ln x)/(1−x)}, and compute the limit to obtain 1/e.

Apply l'Hôpital's rule to the limit 1/x minus cot x as x approaches zero. Rewrite cot x as cos x over sin x and obtain a 0/0 form.

Solve the last L'Hôpital's rule exercise from an infinity over infinity indeterminate form, using chain rule and derivatives of log, sine, and cosine to show the limit equals one.

Explore why polynomials approximate functions, focusing on computational simplicity and the use of Taylor polynomials. Prepare for calculus two topics on Taylor and Maclaurin series and related limits.

Explore how monic monomials near zero influence limits of rational functions, apply L'Hôpital's rule and Taylor polynomials to approximate functions and analyze indeterminate forms.

Review higher order derivatives' existence for Taylor polynomials, clarify order notation with parentheses vs composition, and illustrate C0 through C∞ classes and their strict inclusions with examples.

Construct polynomials that match f's derivatives at x0 to create accurate local approximations. See how derivatives shape the Taylor polynomial and how the remainder controls accuracy, including Maclaurin cases.

Derive the Maclaurin polynomial for e^x using Taylor's theorem, showing derivatives at zero equal one, giving p_n(x)=sum_{k=0}^{n} x^k/k! with remainder vanishing for all x.

Derive the Maclaurin polynomial for sine from derivatives at zero, obtaining an odd-power, alternating-sign series of degree 2n+1, with remainder tending to zero, reflecting sine’s odd-function behavior.

Explore the Maclaurin polynomial for cosine, showing even-powered terms, derivative values at zero, and how the remainder tends to zero for accurate, global approximation.

Explore the second derivative test within Taylor's formula, extend the test to higher derivatives, and understand how the first non-vanishing derivative determines local extrema.

Explore how to approximate functions using a second-degree Taylor polynomial around four to estimate sqrt(4.01), building on linearization and derivative calculations.

Explore computing limits with Taylor's theorem and Maclaurin expansions, using a cusp example, L'Hôpital's rule, and sine and cosine expansions.

Explore the strict order among function classes, using trigonometric examples to illustrate inclusions, discontinuities, continuity at zero, and differentiability across C0, C1, C2, and C infinity.