Overview
This session covered Level 2 Calculus, focusing on five key questions that ranged from achieved to excellence level, addressing differentiation, tangents, turning points, and optimization techniques.
Basic Differentiation and Gradients
- To find the gradient of a function at a point, differentiate and substitute the x-value.
- Example: For ( g(x) = 5x^3 - 4x^2 + 6x - 1 ), ( g'(x) = 15x^2 - 8x + 6 ).
- Substitute ( x = 3 ) to get the gradient at that point.
Tangent to a Curve (Merit Level)
- To prove a line is tangent to a curve at a point, show it passes through the point and has the same gradient as the curve there.
- For ( h(x) = x^2 + 2 ) at ( x=1 ), ( h(1) = 3 ); gradient ( h'(x) = 2x ), so at ( x=1 ), gradient is 2.
- Equation of tangent: ( y - 3 = 2(x-1) ) simplifies to ( y = 2x + 1 ).
Maximum Turning Point & Finding Coefficients
- A turning point means gradient (derivative) is zero.
- For ( g(x) = x^3 - 6x^2 + kx + 4 ), first find ( g'(x) = 3x^2 - 12x + k ).
- If maximum at ( x=1 ), set ( g'(1)=0 ) to solve for ( k ); here, ( k=9 ).
- To find minimum, solve ( g'(x)=0 ) for other values and substitute back to get the y-coordinate.
Using Integration and Given Turning Points (Excellence)
- If given ( f'(x) = 4x - b ) and f has a turning point at (2,7), ( f'(2)=0 ) gives ( b=8 ).
- Integrate to get ( f(x) = 2x^2 - 8x + c ), then substitute ( (2,7) ) to find ( c=15 ).
- Final function: ( f(x) = 2x^2 - 8x + 15 ).
Optimization: Maximum and Minimum Values
- For ( f(x) = 6x^2 - x^3 ), differentiate: ( f'(x) = 12x - 3x^2 ).
- Set ( f'(x)=0 ) to find critical points: ( x=0 ) and ( x=4 ).
- Double differentiate: ( f''(x)=12-6x ); test at each critical point.
- ( f''(0)=12 ) → minimum, ( f''(4)=-12 ) → maximum.
- Substitute ( x = 4 ) into original to get max value: ( f(4) = 32 ).
Key Terms & Definitions
- Gradient — Slope of the tangent to a curve at a point, found via differentiation.
- Turning Point — Where the gradient is zero; can be a maximum or a minimum.
- Tangent — A straight line that touches a curve at exactly one point, matching its gradient there.
- Optimization — Finding maximum or minimum values of functions, often using derivatives.
- Double Differentiation — Differentiating twice to test for maxima or minima: ( f''(x)>0 ) is min, ( f''(x)<0 ) is max.
Action Items / Next Steps
- Review the Level 2 Calculus and Algebra playlists for revision and practice.
- Work through recent exam papers, starting from the latest years and moving backwards.
- Practice finding gradients, writing equations of tangents, and solving optimization problems.
- Prepare questions for the next tutorial or livestream session.