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Find the stationary points on the curve: y = x^3 + 3x^2 +2x+5

Firstly differentiate the function: f(x) = x 3 + 3x 2 + 2x + 5 (function)f'(x) = 3x 2 + 6x + 2 (gradient function) Stationary points are points where the graph has a gradient of zero 3x 2 + 6x + 2 = 0 In ord...
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Answered by Nicolas C. • Maths tutor
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The lines y = 3x² - x + 5/2 intersects the line y = x/2 +7 at two points. Give their coordinates. Show your working

y= 3x 2 -x +5/2 = 0.5 x +76x 2 -2x +5 = x+146x 2 - 3x -9 = 0 = 2x 2 - x -3(2x-3)(x+1) = 0 x = -1 or x = 1.5y = 6.5 or y = 7.75
Answered by • Maths tutor
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A sweet is modelled as a sphere of radius 10mm and is sucked. After five minutes, the radius has decreased to 7mm. The rate of decrease of the radius is inversely proportional to the square of the radius. How long does it take for the sweet to dissolve?

dr/dt propto -1/r^2 and integrate to find equation linking radius and time with boundary conditions. Set r = 0, answer is 7mins 37 seconds.
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Answered by Igor S. • Maths tutor
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Find the area between the curve y = 8 + 2x - x^2 and the line y = 8 - 2x.

First sketch the curve and the line, noting down where they intersect each axis.area under y = 8 + 2x - x 2 is given by the integral between 0 and 4 of (8 + 2x - x 2 ) dx.area under line is given by the inte...
Answered by • Maths tutor
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The rate of growth of a population of micro-organisms is modelled by the equation: dP/dt = 3t^2+6t, where P is the population size at time t hours. Given that P=100 at t=1, find P in terms of t.

First, we integrate the equation with respect to t to find an equation for P. dP/dt = 3t 2 + 6t Then, P= integral (3t 2 + 6t) dt Integrating gives P= t 3 +3t 2 +c, c is the constant of integration. As we are...
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Answered by Claire B. • Maths tutor
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