![]() Two other useful properties concerning inequalities are: $$|a| ≤ b ⇔ -b ≤ a ≤ b$$ $$|a| ≥ b ⇔ a ≤ -b \space or \space b ≤ a$$ Idempotence (the absolute value of the absolute value is the absolute value) $$||a|| = |a|$$ Symmetry $$|-a| = |a|$$ Identity of indiscernibles (equivalent to positive-definiteness) $$|a - b| = 0 ⇔ a = b$$ Triangle inequality (equivalent to subadditivity) $$|a - b| ≤ |a - c| + |c - b|$$ Preservation of division (equivalent to multiplicativeness) $$|a / b| = |a| / |b| \space\space if \space\space b ≠ 0$$ (equivalent to subadditivity) $$|a - b| ≥ ||a| - |b||$$ Other important properties of the absolute value include: The absolute value has the following four fundamental properties: Furthermore, the absolute value of the difference of two real numbers is the distance between them. The absolute value of a number may be thought of as its distance from zero along real number line. The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.įor example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. Solve the Quadratic Equation by the Quadratic Formula.Solve the Quadratic Equation by Factoring.Solve the Quadratic Equation by Extracting Roots.Multiplying and Dividing Positive and Negative Whole Numbers.Finding the value of the function between the x values graphically represents the area of the function under the curve within the x limits. Set the numerator or denominator as different variable (depends on compatibility), differentiate, substitute in appropriate place, rewrite, and then integrate.ĭefinite integral is a basic tool in application of integration. ![]() If dy/dx = x n, then after integration y = x n+1/ n+1 + C, where C is integral constant. Learn the rule of integrating functions and apply it here. Good practice sheets for calculus beginners. Integration worksheets include basic integration of simple functions, integration using power rule, substitution method, definite integrals and more. This printable high school worksheet includes hyperbolic functions such as sinh x, cosh x, tanh x and its reciprocal.Īpply all the rules available and differentiate the functions in terms of y with respect to x. Use this simple technique to solve this worksheet. If y = a f(x), then the derivative of y = a f(x) ln a f '(x). If y = a x, then the derivative of y = a x ln a. Using all necessary rules, solve this differential calculus pdf worksheet based on natural logarithm.ĭifferentiation worksheets based on trigonometry functions such as sine, cosine, tangent, cotangent, secant, cosecant and its inverse. If y = ln x, then the derivative of y = 1/x. ![]() If y = u/v, then the derivative of y = (u'v-uv') / v 2. Quotient rule is a little more complicated than the product rule. If the function y = mn, then the derivative of y = m * derivative of n + n * derivative of m. If y = x n, then the derivative of y = nx n-1.
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