11.01 Antidifferentiation
Consider the function x^{4}.
Calculate \dfrac{d}{dx} \left(x^{4}\right).
Hence find \int 4 x^{3} dx.
Consider the function x^{6}.
Calculate \dfrac{d}{dx} \left(x^{6}\right).
Hence find \int 24 x^{5} dx.
Consider the function x^{5} + x^{4}.
Calculate \dfrac{d}{dx} \left(x^{5} + x^{4}\right).
Find \int \left( 35 x^{4} + 28 x^{3}\right) dx.
Consider the function x^{ - 4 }.
Calculate \dfrac{d}{dx} \left(x^{ - 4 }\right).
Hence find \int 16 x^{ - 5 } dx.
Consider the function x^{8} + x^{6}.
Calculate \dfrac{d}{dx} \left(x^{8} + x^{6}\right).
Find \int \left( 4 x^{7} + 3 x^{5}\right) dx.
Consider the function \sqrt[5]{x^{6}}.
Calculate \dfrac{d}{dx} \left(\sqrt[5]{x^{6}}\right).
Hence find \int 6 \sqrt[5]{x} dx.
Consider the gradient function f \rq \left( x \right) = 2.
Write the general form of the antiderivative function, f \left( x \right).
The gradient function f \rq \left( x \right), is a constant function. What type of function is the antiderivative function, f \left( x \right).
If the form of a constant gradient function is f \rq \left( x \right) = m , write the general form of the antiderivative function.
Consider the gradient function f \rq \left( x \right) = - 8 x.
Write the general form of the antiderivative function, f \left( x \right).
The gradient function f \rq \left( x \right), is a linear function. What type of function is the antiderivative function, f \left( x \right).
If the form of a linear gradient function is f \rq \left( x \right) = ax , write the general form of the antiderivative function.
Consider the gradient function f \rq \left( x \right) = 4 x + 3.
Write the general form of the antiderivative function, f \left( x \right).
The gradient function f \rq \left( x \right), is a linear function. What type of function is the antiderivative function, f \left( x \right).
If the form of a linear gradient function is f \rq \left( x \right) = ax +b , write the general form of the antiderivative function.
Consider the gradient function f \rq \left( x \right) = 7 x^{2}.
Write the general form of the antiderivative function, f \left( x \right).
The gradient function f \rq \left( x \right), is a quadratic function. What type of function is the antiderivative function, f \left( x \right).
If the form of a quadratic gradient function is f \rq \left( x \right) = ax^2 , write the general form of the antiderivative function.
Find the primitive function of the following:
\dfrac{d y}{d x} = 9
\dfrac{d y}{d x} = 8 x
\dfrac{d y}{d x} = 10 x + 7
\dfrac{d y}{d x} = 9 x^{2}
\dfrac{d y}{d x} = 9 x^{2} + 4 x - 6
\dfrac{d y}{d x} = x^{5}
\dfrac{d y}{d x} = 15 x^{4} + 16 x^{3}
\dfrac{d y}{d x} = \dfrac{x^{6}}{4} + \dfrac{x^{2}}{3}
\dfrac{d y}{d x} = x^{ - 6 }
\dfrac{d y}{d x} = \dfrac{15}{x^{6}}
\dfrac{d y}{d x} = \dfrac{10}{x^{6}} - \dfrac{9}{x^{4}}
\dfrac{d y}{d x} = 4 x^{\frac{2}{5}} + 3 x^{\frac{4}{7}}
\dfrac{d y}{d x} = x^{ - \frac{3}{7} } + x^{-\frac{2}{5} }
\dfrac{d y}{d x} = 8 x^{3} + 3 x^{\frac{5}{3}} - 3
\dfrac{d y}{d x} = \dfrac{x^{3} + 4}{x^{3}}
\dfrac{d y}{d x} = x^{2} \left( 10 x^{2} - 9 x\right)
\dfrac{dy}{dx} = \left( 5 x - 2\right) \left( 3 x - 4\right)
\dfrac{dy}{dx} = \left(x + 4\right) \left(x + 6\right)
\dfrac{d y}{d x} = \sqrt{x}
\dfrac{d y}{d x} = 18 \sqrt{x}
\dfrac{d y}{d x} = x \sqrt{x}
\dfrac{d y}{d x} = \dfrac{6}{\sqrt{x}}
For each of the following gradient functions:
State what type of function the antiderivative, f\left(x\right), is.
Find f(x).
Sketch a possible graph for the antiderivative f\left(x\right).
Consider the gradient function f'\left(x\right)=\dfrac{3}{x^{2}}.
Find f\left(x\right).
Sketch a possible graph of the antiderivative f\left(x\right).
The gradient function, f'\left(x\right), has only one x-intercept at \left( - 4 , 0\right), a y-intercept at \left(0, - 3 \right) and a constant gradient.
Find f'\left(x\right).
Find f\left(x\right).
Sketch a possible graph of the antiderivative f\left(x\right).
Consider the equation \dfrac{d y}{d x} = 4 x + 7.
Find a general equation for y.
Find the equation of y, if the curve passes through the point \left(3, 41\right).
Consider the equation \dfrac{d y}{d x} = 9 x^{2} - 10 x + 2.
Find a general equation for y.
Find the equation of y, if the curve passes through the point \left(2, 13\right).
A family of curves has a gradient function y \rq = 15 x^{2} + 7.
Find the equation of y for the family of curves.
Find the equation of the curve that passes through the point \left(2, 59\right).
Consider the gradient function \dfrac{d y}{d x} = 10 x^{4} + 20 x^{3} + 6 x^{2} + 6 x + 9.
Find an equation for y.
Find the equation of y, if the curve passes through the point \left( - 3 , - 133 \right).
Consider the gradient function \dfrac{d y}{d x} = 9 x^{\frac{2}{3}}.
Find an equation for y.
Find the equation of y, if the curve passes through the point \left(8,\dfrac{889}{5}\right).
Find the equation of a curve p in terms of t, given the following:
- \dfrac{d p}{d t} = 6 t - 5
when t = 3, \dfrac{d p}{d t} = 13 and p = 15
Find the equation of a curve y in terms of x, given the following:
- y \rq \rq = 0.
- y contains the points \left(-1, -1 \right) and \left(0, 2 \right).