Learn Calculus I: Beginner to Advanced Quickly

Learn linear functions that grow at a constant rate and form a straight line. Identify the slope and y-intercept, then graph by plotting two points.

Define the slope as the coefficient of x and the y-intercept as the value when x is zero. Learn to find linear equations from two points using the slope formula.

Review polynomials by identifying the leading term and leading coefficient, and determining the degree as the highest power with its coefficient in an example.

Explore rational functions as the ratio p(x)/q(x) and determine domain by excluding x where the denominator equals zero, such as x ≠ 5 for q(x) = x^2 - 25.

Explore algebraic functions, including polynomials and rational functions, and analyze where they are defined and undefined on the real numbers, with examples of domain restrictions.

Explore sine, cosine, and tangent defined through right triangles and angle ratios; study their graphs and domain and range relationships, including sine and cosine between -1 and 1.

examine exponential functions of the form f(x)=b^x, where b>0 and b≠1, noting that b>1 increases the graph while 0<b<1 decreases it, with domain all real numbers and positive range.

Explore logarithmic functions and how to find the exponent to reach a target value using log base a, and analyze the exponential graph y = a^x, including domain and range.

Explore intercepts of a graph by identifying x- and y-intercepts, solving for x when y=0, and applying examples like y=3x-5 and y=x^2-2.

Explore even and odd functions by testing f(-x) against f(x) and -f(x), illustrating y-axis and origin symmetry with examples like x^2+1 and 2x.

Explore vertical transformations of a function by adding or subtracting constants to its output, moving the graph up or down, with examples like g(x)=f(x)+2 and g(x)=f(x)-3.

Learn horizontal shifting by adjusting x to move a graph left or right; subtract from x to shift right, add to x to shift left, with f(x)=x^2 as an example.

An instructor demonstrates shifting the graph of f(x) = x^3 by vertical and horizontal translations, moving up 2 units with f(x) + 2 and left 2 units with f(x+2).

Multiply a function by constants to vertically stretch or compress its graph, illustrated with cosine of x, showing stretch by 2 or 3 and compression by 1/2.

Learn how to horizontally stretch or compress a function by multiplying x by a constant. Recognize that constants greater than one compress the function; those less than one stretch it.

Learn how to reflect functions vertically by multiplying by -1 and horizontally by replacing x with -x, using examples like sqrt(x) and x^2 to show domain and graph changes.

Explain piecewise functions defined by three subfunctions using doctor’s fee: 50 up to 5 minutes, 80 5–15, 6 per minute beyond 15; 7 minutes = 80, 20 minutes = 120.

Learn how to test inverse functions by composing them and by plugging one into the other, and find inverses by swapping x and y and solving for y.

Explains the composition of functions by applying g to x and then applying f to that result, as in f(g(2)) = 11.

explain how to compute f∘g and g∘f for f(x)=4x-2 and g(x)=x^2, including domain considerations when square roots are involved.

Determine the derivative of constants using the constant rule, which states a constant's derivative with respect to x is zero, illustrated with simple examples.

Apply the constant multiple rule to pull out constants when differentiating functions, then use the power rule to differentiate x^3 and 9x, yielding 15x^2 and 9 respectively.

Apply the power rule to differentiate x^n by calculating n x^{n-1}, as shown with x^2, x^5, -9x^2, and sqrt(x) via x^{1/2}.

Learn to differentiate sums and differences of functions using the sum and difference rules, with examples involving sine, cosine, tangent, and polynomial terms such as x^2 and x^4.

Learn how to differentiate products using the product rule. Apply the constant multiple rule and the power rule in examples.

Learn how to apply the quotient rule to derivatives of a quotient of two functions, using f over g, with example derivative calculations.