lecture 11 differentiation part 2 ♡

Determine the intervals for which the function $f (x) = \frac{x ^{2}}{x ^{2} + 3 x - 4}$ has positive gradient.

To have positive gradient, f’(x) is positive.

$f^{'} (x) = \frac{( 2 x ) ( x ^{2} + 3 x - 4 ) - ( x ^{2} ) ( 2 x + 3 )}{x ^{4}}$ given $x^{2} + 3 x - 4 \neq = 0$

Find the point of inflection of $f (x) = x^{3} + 6 x^{2} + x - 1$ .

$f^{'} (x) = 3 x^{2} + 12 x + 1$
$f^{''} (x) = 6 x + 12$

At SPOI, $f^{''} (x) = 0$

$0 = 6 x + 12$
$x = - 2$
$f (- 2) = 13$

$(- 2, 13)$

The tangent to the curve of $y = a x^{3} - b x + 5$ at the point $x = 1$ is parallel to the line $y = 10 x$ . The curve passes through $(- 1, 3)$ . Find a and b.

let $f (x) = y$
$10 = f^{'} (1)$
$f^{'} (x) = 3 a x^{2} - b$
$f^{'} (1) = 3 a - b = 10$
$f (- 1) = 3 = - a + b + 5$
$a - b = 2$
$2 a = 8$
$a = 4$
$b = 2$

Let $f : (- \infty, \frac{1}{2}] \to R, f (x) = 1 - 2 x$ . Find the positive angle $θ$ from the positive x axis to the tangent at $x = - 1$ .

$f^{'} (x) = \frac{1}{2} (1 - 2 x)^{\frac{- 1}{2}} (- 2)$
$f^{'} (x) = - (1 - 2 x)^{\frac{- 1}{2}}$
$f^{'} (- 1) = tan (θ)$
$f^{'} (- 1) = \frac{- 1}{3}$
$θ = \frac{5 π}{6}$

Let $f : [1, 4) \to R, f (x) = - x^{2} + 2 x - 1)$ . Find the absolute maximum using differentiation.

$f^{'} (x) = - 2 x + 2$
$- 2 x + 2 = 0$
$x = 1$
$f (1) = 0$

The absolute maximum is 0.

Let $f (x) = (x - 3)^{2} - 4$ . At what value of x is the tangent to the curve parallel to the line connecting its smaller x-intercept and the y-intercept?

Graphing is important.

First find the x-intercepts.

$(x - 3)^{2} - 4 = 0$
$x - 3 = \pm 2$
$x = 1, 5$

For $f (x) = \frac{1}{2 x - 1}$ how many stationary points are there?

Stationary point means $f^{'} (x) = 0$

$f^{'} (x) = 0 = \frac{- 2}{( 2 x - 1 ) ^{2}}$
$- 2 = 0$

Therefore there are no stationary points.

A cylindrical can is to be constructed using a maximum of 1000 cm2 of aluminium and its volume maximised. What is the radius?

$S = 2 π r^{2} + 2 π r h = 1000$
$V = hπ r^{2}$
$h = \frac{1000 - 2 π r ^{2}}{2 π r}$
$V = π r^{2} (\frac{1000 - 2 π r ^{2}}{2 π r})$
$V = \frac{r ( 1000 - 2 π r ^{2} )}{2}$

$V = \frac{1000 r - 2 π r ^{3}}{2}$

We want the maximum, derivative is zero.

$\frac{d V}{d r} = 0 = \frac{1000 - 6 π r ^{2}}{2}$

$6 π r^{2} = 1000$
$r = (\frac{500}{3 π})^{\frac{1}{2}}$

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lecture 11 differentiation part 2

Determine the intervals for which the function $f (x) = \frac{x ^{2}}{x ^{2} + 3 x - 4}$ has positive gradient.

Find the point of inflection of $f (x) = x^{3} + 6 x^{2} + x - 1$ .

The tangent to the curve of $y = a x^{3} - b x + 5$ at the point $x = 1$ is parallel to the line $y = 10 x$ . The curve passes through $(- 1, 3)$ . Find a and b.

Let $f : (- \infty, \frac{1}{2}] \to R, f (x) = 1 - 2 x$ . Find the positive angle $θ$ from the positive x axis to the tangent at $x = - 1$ .

Let $f : [1, 4) \to R, f (x) = - x^{2} + 2 x - 1)$ . Find the absolute maximum using differentiation.

Let $f (x) = (x - 3)^{2} - 4$ . At what value of x is the tangent to the curve parallel to the line connecting its smaller x-intercept and the y-intercept?

For $f (x) = \frac{1}{2 x - 1}$ how many stationary points are there?

A cylindrical can is to be constructed using a maximum of 1000 cm2 of aluminium and its volume maximised. What is the radius?

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wiki ♡

Explorer

lecture 11 differentiation part 2

Determine the intervals for which the function f(x)=x2+3x−4x2​ has positive gradient.

Find the point of inflection of f(x)=x3+6x2+x−1.

The tangent to the curve of y=ax3−bx+5 at the point x=1 is parallel to the line y=10x. The curve passes through (−1,3). Find a and b.

Let f:(−∞,21​]→R,f(x)=1−2x​. Find the positive angle θ from the positive x axis to the tangent at x=−1.

Let f:[1,4)→R,f(x)=−x2+2x−1). Find the absolute maximum using differentiation.

Let f(x)=(x−3)2−4. At what value of x is the tangent to the curve parallel to the line connecting its smaller x-intercept and the y-intercept?

For f(x)=2x−11​ how many stationary points are there?

A cylindrical can is to be constructed using a maximum of 1000 cm2 of aluminium and its volume maximised. What is the radius?

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Table of Contents

Determine the intervals for which the function $f (x) = \frac{x ^{2}}{x ^{2} + 3 x - 4}$ has positive gradient.

Find the point of inflection of $f (x) = x^{3} + 6 x^{2} + x - 1$ .

The tangent to the curve of $y = a x^{3} - b x + 5$ at the point $x = 1$ is parallel to the line $y = 10 x$ . The curve passes through $(- 1, 3)$ . Find a and b.

Let $f : (- \infty, \frac{1}{2}] \to R, f (x) = 1 - 2 x$ . Find the positive angle $θ$ from the positive x axis to the tangent at $x = - 1$ .

Let $f : [1, 4) \to R, f (x) = - x^{2} + 2 x - 1)$ . Find the absolute maximum using differentiation.

Let $f (x) = (x - 3)^{2} - 4$ . At what value of x is the tangent to the curve parallel to the line connecting its smaller x-intercept and the y-intercept?

For $f (x) = \frac{1}{2 x - 1}$ how many stationary points are there?