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Find the exact solution of the equation in its simplest form: 3^x * e^4x = e^7.

First take natural logs of both sides, giving us: ln(3 x * e 4x ) = ln(e 7 ). The RHS can be simplified using the definition of natural logs, so ln(e 7 ) = 7. We can then apply the product rule for logs (ln(...
HM
Answered by Hugo M. • Maths tutor
5879 Views

Express x^2-7x+2 in the form (x-p)^2+q where p and q are rational. Hence or otherwise find the minimum value of x^2-7x+2

For the first part, use the 'complete the square method', which gives (x-7/2) 2 -49/4+2, simplified to (x-7/2) 2 -41/4. To find the answer for the second part, set x=7/2 so that the bracketed part becomes 0,...
CW
Answered by Connaire W. • Maths tutor
5302 Views

Find the binomial expansion of (2+x)^3

We know thst the binomial expansion formula is (1+x) n = 1+nx +(n)(n-1)(x 2 )/2! ...etc Thus we want to rearrange the expression we are given into the form (1+x) n so that we can apply the formula to it. Tak...
LV
Answered by Laura V. • Maths tutor
11323 Views

Given that y = 3x(^2) + 6x(^1/3) + (2x(^3) - 7)/(3(sqrt(x))) when x > 0 find dy/dx

Firstly, the (2x(^3) - 7)/(3(sqrt(x))) can be split into (2x(^3))/(3(sqrt(x)) and -7/(3(sqrt(x)). These can then be simplified to (2/3)x(^5/2) and -(7/3)x(^-1/2) respectively. This then gives the equation y ...
SH
Answered by Samuel H. • Maths tutor
17608 Views

Integrate, by parts, y=xln(x),

First, we need to separate the RHS as components of U and V. Using the LATE (logarithms, algebra, trigonometry, exponentials) technique, we see that logarithms have priority to algebra hence U = lnx and dV/d...
MS
Answered by Makhdoom S. • Maths tutor
3761 Views