For the function `f:R→R,\ \ f(x)= 2e^x + 3x`, determine the coordinates of the point `P` at which the tangent to `f(x)` is parallel to the line `y = 5x - 3`. (3 marks)
Calculus, MET2 2016 VCAA 2
Consider the function `f(x) = -1/3 (x + 2) (x - 1)^2.`
- i. Given that `g prime (x) = f (x) and g (0) = 1`, show that `g(x) = -x^4/12 + x^2/2 - (2x)/3 + 1`. (1 mark)
- ii. Find the values of `x` for which the graph of `y = g(x)` has a stationary point. (1 mark)
The diagram below shows part of the graph of `y = g(x)`, the tangent to the graph at `x = 2` and a straight line drawn perpendicular to the tangent to the graph at `x = 2`. The equation of the tangent at the point `A` with coordinates `(2, g(2))` is `y = 3 - (4x)/3`.
The tangent cuts the `y`-axis at `B`. The line perpendicular to the tangent cuts the `y`-axis at `C`.
- i. Find the coordinates of `B`. (1 mark)
- ii. Find the equation of the line that passes through `A` and `C` and, hence, find the coordinates of `C`. (2 marks)
- iii. Find the area of triangle `ABC`. (2 marks)
- The tangent at `D` is parallel to the tangent at `A`. It intersects the line passing through `A` and `C` at `E`.
- ii. Find the length of `AE`. (3 marks)
Calculus, MET2 2016 VCAA 10 MC
For the curve `y = x^2 - 5`, the tangent to the curve will be parallel to the line connecting the positive x-intercept and the y-intercept when `x` is equal to
A. `sqrt 5`
B. `5`
C. `−5`
D. `sqrt 5/2`
E. `1/sqrt 5`
Calculus, MET1 2015 VCAA 10
The diagram below shows a point, `T`, on a circle. The circle has radius 2 and centre at the point `C` with coordinates `(2, 0)`. The angle `ECT` is `theta`, where `0 < theta <= pi/2`.
The diagram also shows the tangent to the circle at `T`. This tangent is perpendicular to `CT` and intersects the `x`-axis at point `X` and the `y`-axis at point `Y`.
- Find the coordinates of `T` in terms of `theta`. (1 mark)
- Find the gradient of the tangent to the circle at `T` in terms of `theta`. (1 mark)
- The equation of the tangent to the circle at `T` can be expressed as
- `qquad cos(theta)x + sin(theta)y = 2 + 2cos(theta)`
- i. Point `B`, with coordinates `(2, b)`, is on the line segment `XY`.
- Find `b` in terms of `theta`. (1 mark)
- ii. Point `D`, with coordinates `(4, d)`, is on the line segment `XY`.
- Find `d` in terms of `theta`. (1 mark)
- Consider the trapezium `CEDB` with parallel sides of length `b` and `d`.
- Find the value of `theta` for which the area of the trapezium `CEDB` is a minimum. Also find the minimum value of the area. (3 marks)