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A Polynomial is defined as X^3-6X^2+11X-6. a)i Use the factor theorem to show that X-3 is a factor. ii Express as a linear and quadratic b)Find the first and second derivative c) Prove there is a maximum at y=0.385 to 3DP

a)i Factor Theorem states that if a polynomial has a factor of X-a, when a is substituted in, the polynomial = 0. Therefore substitute 3 into the polynomial, which then equals 0 ii Long Devision- Unsure how ...
OM
Answered by Oliver M. • Maths tutor
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Using the limit definition of the derivative, find the derivative of f(x)=sin(3x) at x=2π

First, we need to find the derivative f'(x) for all x, and then we will substitute x=2π into the function we find. To do this, we need to set up our limit: f'(x) = lim [sin(3(x+h))-sin(3x)]/h as h-> 0 We ...
Answered by • Maths tutor
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Show by induction that sum_n(r*3^(r-1))=1/4+(3^n/4)*(2n-1) for n>0

Base Case n=1sum_1(r 3^(r-1))=1 3^0=1=1/4+(3^1/4) (2-1)=1/4+3/4=1 Assume true for n=ki.e. sum_k(r 3^(r-1))=(3^k/4) (2k-1) then for n=k+1sum_(k+1)(r 3^(r-1))= sum_k(r 3^(r-1))+(k+1) 3^k =(3^k/4) (2k-1)+(k+1) ...
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Find the stationary points of y = 4(x^2 - 4)^3

This question challenges the student's ability to use the chain rule and to understand what a stationary point is. Having recognised that the chain rule is required, the first step is to use the substitution...
TL
Answered by Tom L. • Maths tutor
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Calculate the integral of ln(x)

Calculating the integral of ln(x) is harder than it might look and must be done using integration by parts, where the two parts are 1 and ln(x). The integration by parts formula is as follows integral{udv) =...
CR
Answered by Callum R. • Maths tutor
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