question 7: differential calculus
this one is all about algebra, pay attention to your algebra and you will be just fine
7.1
- derivative from first principles is usually free marks and everyone usually gets them.
- Your job is to substitute x+h on x of the given equation and subtract that equation
- divide by h, factorise h on the numerator in order to eliminate the denominator
- substitute zero on the remaining h
7.2
- use power rule to derive, but before that, you need exponents
- use exponents and surds rules to change square root and negative exponents
- then drop the exponent to multiply the co-efficient, subtract 1 on the exponent
7.3
- once again, we use power rule to derive, keep the same notation and remove brackets on final answer
- we are deriving t, since the notation says dt of, therefore g is a constant
solution:
question 8: cubic function
key concepts:
- x-intercepts: equate the equation to zero and use synthetic division to solve for x
- y-intercept: substitute x with zero, or take the constant
- stationary points: these are the two turning points, find first derivative, equate to zero and solve for x, sub in the original equation to find y
- point of inflection: the point where concavity changes, second derivative equal to zero and solve for x
- tangent to graph: the first derivative describes gradient at any point of the graph, gradient at stationary point is zero (since there’s no change in y), that’s why we equate to zero when solving stationary points. Equate to given gradient when solving other points.
8.1
- c is the y-intercept and is (0;9), the constant
8.2
- x-intercepts explained above
8.3
- concavity changes at point of inflection, where second derivative is equal to zero
- we must find that point and decide
- in this case, the graph is concave down on the left side of point of inflection, so x will be less than the point of inflection.
8.4
- the second derivative of a cubic function gives us a parabolic function
- so, we determine the x-intercepts of a parabola in order to find out through inspection where the graph is below or above the x-axis.
- Since the parabola will be positive here, because the cubic is also positive, the graph will be below the x-axis from turning point until x-intercepts.
- So, x is between the two intercepts
Solution:
