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131 results.

Solving equations
Draft
Question in Bill's workspace by Bill Foster

Solve for $x$ : $\displaystyle ax ^ 2 + bx + c=0$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $(ax+b)^m(cx+d)^n$ using the product rule. The answer will be of the form $(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$ . Find $g(x)$ .
Quotient rule
Draft
Question in Bill's workspace by Bill Foster

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$ . Find $g(x)$ .
Quotient rule
Draft
Question in Bill's workspace by Bill Foster

The derivative of $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$ . Find $g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $\sin(ax+b) e ^ {nx}$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $x^m\cos(ax+b)$
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $(a+bx) ^ {m} \sin(nx)$
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $x ^ m(ax+b)^n$ using the product rule. The answer will be of the form $x^{m-1}(ax+b)^{n-1}g(x)$ for a polynomial $g(x)$ . Find $g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate the function $(a + b x)^m e ^ {n x}$ using the product rule.
Product Rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate the function $f(x)=(a + b x)^m e ^ {n x}$ using the product rule. Find $g(x)$ such that $f\;'(x)= (a + b x)^{m-1} e ^ {n x}g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $f(x) = x^m(a x+b)^n$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $x ^m \sqrt{a x+b}$ .
The answer is in the form $\displaystyle \frac{x^{m-1}g(x)}{2\sqrt{ax+b}}$
for a polynomial $g(x)$ . Find $g(x)$ .
Complete the square. (Video)
Draft
Question in Bill's workspace by Bill Foster

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$ .

Hence, or otherwise, find roots of $ax^2+bx+c=0$ .

Includes a video which shows how to solve a quadratic by completing the square.
Chain rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $\displaystyle \ln((ax+b)^{m})$
Chain rule. (Video).
Draft
Question in Bill's workspace by Bill Foster

Differentiate $\displaystyle \cos(e^{ax}+bx^2+c)$ .

Contains a video solving a similar example.
Chain rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $\displaystyle e^{ax^{m} +bx^2+c}$
Chain rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate

$\sqrt{a x^m+b})$
Factorise a quadratic
Draft
Question in Bill's workspace by Bill Foster

Factorise $\displaystyle{ax ^ 2 + bx + c}$ into linear factors.
Complete the square
Draft
Question in Bill's workspace by Bill Foster

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$ .
Integration by substitution
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.
Interactive Newton-Raphson method
Should not be used
Question in Christian's workspace by Christian Lawson-Perfect and 1 other

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$ . If $x_0=1$ find $x_1$ .

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.
Cartesian equation of a plane
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$ .

There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx$ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx$
Indefinite integral by substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$ . Solution involves the inverse trigonometric function $\arcsin$ .
Calculate definite integral with a hyperbolic substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$
Integration: Indefinite integral
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$ .
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$ , find $g(x)$ . Find $h(x)$ , $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$ .
Integration: Indefinite integral by substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$
Integral by hyperbolic substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$
Integration: Indefinite integral
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

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