Investigating shapes of graphs
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To find the maximum and minimum points on a graph y=f(x): 1. Find the expression for the derivative f'(x) and set it equal to 0 to find stationary points. 2. Check that at these points, f'(x) is negative on the left and positive on the right for a minimum, or positive on the left and negative on the right for a maximum. 3. The y-values at these x-values give the minimum or maximum points. The second derivative test can also determine if a stationary point is a maximum or minimum.
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This document defines polynomials and discusses their key properties. It begins by defining a polynomial as an algebraic expression with two or more terms where the power of each variable is a positive integer. The degree of a polynomial is defined as the highest power of the variable. Polynomials are then classified based on their degree as constant, linear, quadratic, cubic, etc. The document also discusses the zeros or roots of a polynomial, which are the values that make the polynomial equal to zero. It shows how the zeros relate to the coefficients of the polynomial and can be found using factoring or solving techniques. Examples are provided to illustrate dividing polynomials using the division algorithm.
Presentaton on Polynomials....Class 10
This document discusses various concepts related to polynomials including: the degree of polynomials such as constant, linear, quadratic, and biquadratic polynomials; the value and zeros of polynomials; the relationship between the coefficients and zeros of polynomials; and the division algorithm for polynomials. Key points covered include that the degree of a polynomial refers to its highest exponent, the graph of polynomials results in various curve shapes depending on degree, and that the coefficients and zeros of polynomials are related through properties like the sum and product of the zeros.
differentiate free
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
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Hari kemerdekaan is the independence day of malaysia
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Investigating shapes of graphs
- 1. Investigating shapes of graphs Maximum and minimum points On a graph y = f(x), there is a maximum at x = q if f(x) is less than f(q) on either side of x=q. For this to be the case then f’(x) has to be positive to the left of q, negative to the right of q, and zero at x = q. Similarly there to be a minimum point at x = q then f’(q)=0, with f’(x) < 0 for x < q and f’(x) > 0 for x > q. Values where f’(x) = 0 are called stationary points or turning points To find the maximum and minimum points on the graph y = f(x) 1. Find an expression for )(' xf dx dy = 2. List the values of x for which 0= dx dy 3. Taking these values in turn find the sign of dx dy to the left and right of that value. 4. If the values are – and + respectively then it is a minimum point. If they are + and – respectively then it is a maximum point. 5. For each value of x which gives a stationary point find the corresponding value of y = f(x) Second derivatives If we take the derivative of f’(x) we get a ‘second derivative’, 2 2 dx yd . For a stationary point if 2 2 dx yd <0 then it is maximum, whilst if 2 2 dx yd >0 it is a minimum