@?further-maths

See Also [[Maths - Differentiation]]S https://betterexplained.com/articles/derivatives-product-power-chain/ Flashcards What is the chain rule?? $$ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} $$ What new variable do you introduce when using the chain rule?? $$ u $$ When do you apply the chain rule?? When you have composite functions. $$y = (3x + 4)^5$$ What substitution would you make in order to differentiate?? $$ u = 3x + 4 \\ y = u^5 $$ $$y = u^5$$ What is $\frac{dy}{du}$?

See Also [[Maths - Differentiation]]S Flashcards $$y = a^x \ y = a^{-x}$$ What is true about these two graphs?? They are reflections of each other in the $y$-axis. $$y = a^x$$ What is the $y$-intercept of this graph?? $$ 1 $$ $$\log_a b = c$$ If this is true, what is also true?? $$ a^c = b $$ $$3^x = 9$$ What would you do to both sides to make $x$ the subject?

What is the simple case of partial fractions?? Where the denominator is $(ax + b)(cx + d)$. What is the harder case of partial fractions?? Where the denominator is $(ax + b)(cx + d)^2$ $$\frac{5x}{(x + 2)(x - 3)}$$ What’s the first step to finding the partial fractions?? Rewriting as $$ \frac{A}{x + 2} + \frac{B}{x - 3} $$ $$\frac{A}{x + 2} + \frac{B}{x - 3}$$ How could you add these two fractions together?? $$ \frac{A(x - 3) + B(x + 2)}{(x+2)(x-3)} $$

See Also [[Further Maths - Series]]S Flashcards What is the requirement of a series for the method of differences to be applicable?? The general term, $u_r$ of a series can be expressed in the form $f(r) - f(r+1)$. If the general term of a series $u_r$ can be expressed as $f(r) - f(r + 1)$, how could you write the series?? $$ \sum^n_{r = 1} (f(r) - f(r + 1)) $$

2021-01-14 What is the word explanation for the scalar/dot produt of two vectors?? The sum of the products of the components. What’s the notation for the dot product of $\pmb{a}$ and $\pmb{b}$?? $$ \pmb{a} \cdot \pmb{b} $$ What’s the sum formula for $\pmb{a} \cdot \pmb{b}$?? $$ \sum \pmb{a}_i \pmb{b}_i $$ $$ \left(\begin{matrix} 2 \ 2 \ 2 \end{matrix}\right) \cdot \left(\begin{matrix} 1 \ 2 \ 3 \end{matrix}\right) $$ What is the dot product of the two vectors?

2021-01-14 What is the general vector equation of a plane?? $$ \pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c} $$ $$\pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c}$$ What must be true about the two directional vectors $\pmb{b}$ and $\pmb{c}$?? They are not parallel to one another. What equation does this photo represent?? $$ \pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c} $$ $$ \left(\begin{matrix} 3+2\lambda+\mu \ 4+\lambda-\mu \ -2+\lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} 2 \ 2 \ -1 \end{matrix}\right)$$ How could you rewrite this?

2021-01-11 What is the vector equation for a straight line?? $$ \pmb{r} =\pmb{a} + \lambda \pmb{b} $$ $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the variable $\pmb{a}$ called?? The position vector. $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the variable $\pmb{b}$ called?? The direction vector. $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the variable $\pmb{r}$ called?? The general position vector for a point on the line. $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the informal name for $\lambda$ called here?

See also: [[Further Maths - Vector Equation of a Line]]S [[Further Maths - Vector Equation of a Plane]]S [[Further Maths - Dot Product]]S What is the formula for the distance to a three dimensional point $(a,b,c)$?? $$ \sqrt{a^2 + b^2 + c^2} $$ $$ \left(\begin{matrix} a \ b \ c \end{matrix}\right) $$ What is the formula for the length of the vector?? $$ \sqrt{a^2 + b^2 + c^2} $$

What is the general technique for proving a statement about matricies using induction?? Multiplying out the matrix for $k+1$ and showing that it’s the same as substituting $k = k+1$ into the normal $k$ matrix. How could you rewrite $M^{k+1}$?? $$ M^k M $$ Backlinks [[Further Maths - Syllabus]]S Metadata date: 2021-01-05 12:26 tags: - '@?further-maths' - '@?induction' - '@?public' title: Further Maths - Induction for Matricies

If you add a multiple of $4$ to something already divisible by $4$, what must be true about the answer?? It is also divisible by 4. What is the two-step general technique used to show divisibility in induction?? Assume $f(k)$ is divisible Show the difference between $f(k)$ and $f(k+1)$ is divisible. How would you write the difference between $f(k)$ and $f(k+1)$?? $$ f(k+1) - f(k) $$ If $$f(k + 1) - f(k) = 4\times 3^{2k}$$, what other statement could you write that shows clearly $f(k+1)$ is divisible by $4$?

What is $$\sum^{1}_{r = 1} r$$?? $$ 1 $$ What do you get if you substitute $k = 1$ for $\frac{1}{2}k{k+1}$?? $$ 1 $$ How could you write out the sum that is being done for $$\sum^{k}_{r = 1} r$$?? $$ 1 + 2 + 3 + … + (k - 1) + k $$ How could you write out the sum that is being done for $$\sum^{k + 1}_{r = 1} r$$?? $$ 1 + 2 + 3 + … + k + (k + 1) $$

See also: [[Further Maths - Induction for Series]]S [[Further Maths - Induction for Divisibility]]S What is induction?? A proof technique that shows a statement is true for natural numbers. How could you use induction to prove you can climb as high as you like on a ladder?? You can climb onto the bottom rung You can climb onto the next rung What are the 4 steps for induction?