Exercises: Differentiating Polynomials and Tangent Lines

Differentiate each of the following with respect to $x$:

1. a) $f(x) = x^3$

1. b) $f(x) = 4x^2 - 5x$

1. c) $f(x) = 2x^3 - 3x^2 + 7x - 9$

Given $f(x) = x^3 - 6x$, find:

1. a) $f'(x)$

1. b) $f'(2)$

1. c) $f'(-1)$

Find the slope of the curve $y = x^2 + 3x$ at the point where $x = 4$.

The curve $y = x^3 - 3x^2 + 2$ is given.

1. a) Find $\dfrac{dy}{dx}$.

1. b) Find the equation of the tangent to the curve at the point where $x = 2$.

Find the coordinates of the points on the curve $y = x^2 - 6x + 5$ at which the slope of the tangent is $4$.

A ball is thrown upwards from a cliff near Moher. Its height above the sea, in metres, after $t$ seconds is $h(t) = 60 + 20t - 5t^2$.

1. a) Find the speed $h'(t)$ at time $t$.

1. b) Find the time at which the ball is momentarily at rest.

1. c) Find the maximum height reached.

The curve $f(x) = x^3 - 4x^2 + 5x - 2$ is given.

1. a) Find the coordinates of the two points on the curve at which the tangent is parallel to the $x$-axis.

1. b) Find the equation of the tangent to the curve at the point where $x = 0$.

1. c) Find where this tangent meets the curve again.

The line $y = 5x - 4$ is a tangent to the curve $y = x^2 + bx + c$ at the point $(2, 6)$.

1. a) Use the point $(2,6)$ to write an equation connecting $b$ and $c$.

1. b) Use the slope of the tangent to find another equation connecting $b$ and $c$.

1. c) Hence find the values of $b$ and $c$.

A farmer has $120$ m of fencing to make a rectangular pen against a long straight wall, so only three sides need fencing. The two sides perpendicular to the wall each have length $x$ metres.

1. a) Show that the area is $A = 120x - 2x^2$.

1. b) Find $\dfrac{dA}{dx}$ and hence find the value of $x$ that gives the maximum area.

1. c) Find the maximum area and justify that it is a maximum.