Calculus Problem Solving Techniques and Formulas

Taylor Polynomials and Derivatives

Start by finding the derivatives of f(x) as needed. In the example question, it wants the degree 3, so we calculate up to the 3rd derivative, f'''(x), and then use the formula.

Solving Differential Equations

The equation ds/dt = sin(t) shows us that it is a pure time differential, meaning taking the integral of ds/dt will get us s(t). Since we know s(0) = 0, we use this to find the constant C. Once found, substitute it into s(3), and you have solved the problem.

Euler's Method for Numerical Solutions

To use Euler's Method, create your table with x moving at the rate given as Δ. Use the initial points and the formula provided. If given a function f(x), find the derivative. If already given a derivative or dy/dx, keep it there and input your points to find the slope. Make a line equation (see back). Then, increase the x value (or equation) by the Δ given and put it into the new line equation to find y. These are your new points. Input the new points into dy/dx and repeat until finished.

Definite Integrals and Area Calculation

Remember to apply both the top and bottom limits to the formula, and subtract the bottom from the top.

Riemann Sums and Endpoint Methods

Riemann Sums: First, find n. n equals the amount of sections we will be breaking our area into. Divide your area by this number to find the width of each section. Starting from your beginning point (the bottom number on the integral), add your width up until your final point (the top number of the integral).

• Right Endpoints: If finding the right endpoint, do not use width 0; in the example's case, we would not use 0. The right endpoint moves up, then right to the line.
• Left Endpoints: In left endpoints, we do not use the last width; in the example's case, this is π. The left endpoint moves up, then left to the line.

Geometry and Algebra Fundamentals

Point-Slope Form: y - Y = (slope)(x - X). The capital letters represent your points.

Cobwebbing: Move horizontally from the updating function to y = x, then move vertically.

Derivative and Integration Rules

• Derivative Rules: a'b + ab' (Product Rule), (a'b - ab') / b² (Quotient Rule), and outside > inside (Chain Rule).
• Integral Rules: Integration by parts (uv - ∫v du) and substitution method.

Analyzing Functions: Extrema and Critical Points

Use the First Derivative Test to find local maxima and minima by using numbers around critical points. The Second Derivative Test uses the critical points themselves.

Critical Points: These occur where the derivative equals 0 or is undefined.

Implicit Differentiation and Linear Approximation

Implicit Differentiation: Apply dy/dx (or the equation) and numbers, then solve.

Linear Approximation: Take the derivative of the line and apply the given x (not the number you are attempting to approximate, but the one close to it) to find the slope. Put this into point-slope form with the points given (*). Then, the number you are approximating goes into your x variable; solve for y.