Step by Step Calculus 1 : Learn from scratch

Explore limits and velocity concepts by analyzing a car's motion: from a function of time to average velocity to instantaneous velocity via the limit of time intervals.

examine left and right limits at six for a piecewise function, determine when the limit exists, and show how shifting the function can align the sides to a common value.

Examine how the squeeze theorem determines limits by bounding f(x) between g(x) and h(x) near a, then equating their limits to l.

Explore the three types of continuity—including removable and infinite discontinuities—through piecewise functions, limit analysis, and visual methods, with Gaussian and 1/x examples.

Explore lines and slopes, learn the line equation y = mx + b, and identify x-intercepts and y-intercepts while understanding slope as rise over run and its link to derivatives.

Explore the derivative at a point by constructing tangent lines that touch the curve at a single point, and use secant slopes and limits to find the tangent slope.

Learn to differentiate constant functions and power functions using the limit definition, and apply the power rule d/dx x^n = n x^{n-1}.

Derive that the derivative of a constant times a function equals the constant times the derivative, using the definition, with a basic problem and an introduction to the power rule.

Apply the rule that the derivative of a constant times a function equals the constant times the derivative. Differentiate 5x^3 + 7x^9 - x + 1 to get 15x^2 + 63x^8 - 1.

Explore the distribution property of the derivative by differentiating h(x) = g(x) + f(x) + c(x) and showing the derivative equals g'(x) + f'(x) + c'(x).

Learn step-by-step how to differentiate quotients using the quotient rule, with f(x)=x^2 and g(x)=x^3−2, deriving (f/g)' = (f' g − f g')/g^2 and applying it to an example.

practice problem demonstrates applying the quotient rule to differentiate f(x)/g(x) with g(x) = 5 sqrt(x) and f(x) involving a square root.

Derive the derivatives of sine and cosine from the limit definition, showing d/dx sin x = cos x and d/dx cos x = -sin x, and review derivative notations.

Learn how the product rule derives the derivative of f(x)g(x) by f'(x)g(x) + f(x)g'(x). See examples with x^2 and cos x and extend to three or more factors.

Apply the chain rule to derivatives of composite functions by differentiating the outer function and multiplying by the inner derivative. Use a cosine example with g(x)=x^3−x^2+5 to compute derivatives.

Apply the chain rule to differentiate a composite function with the outer tangent and inner three times the square root of x, multiplying derivatives.

Explore implicit differentiation and learn to differentiate with respect to x and y, applying the quotient rule to reveal how x and y relate in implicit equations.

Explore rates of change by linking slope and derivative. Relate volume growth to radius change for a sphere using implicit differentiation.

Solve a rates of change problem where two objects start together, moving north at 5 m/min and east at 12 m/min, using s = sqrt(y^2 + x^2) to find ds/dt.

Explores locating inflection points via the second derivative, deriving f''(x)=6x to find x=0 and the inflection point at (0,0), and examines how tangent slopes change across it.

Identify the maximum and minimum points and how the slope changes around them. Then classify convex or concave cavities near x = sqrt(4/3) and craft the function.

Practice problem sketches the graph of x^3-4x by locating intersection points. It identifies maxima, minima, and inflection points and describes convex and concave regions before sketching.