MYTUTOR SUBJECT ANSWERS

249 views

Find the gradient of a curve whose parametric equations are x=t^2/2+1 and y=t/4-1 when t=2

Remember that the gradient of a curve is expressed as dy/dx. This can be solved by using the chain rule:

dy/dx = dy/dt*dt/dx. The dt in the denominator of the first term, and the numerator of the second term will cancel. It is also useful to remember that dt/dx is the same as 1/(dx/dt).

Now all we have to do to solve this is find the differential with respect to t of the the two parametric equations. Remember that if we have an equation where there is more than one term (i.e. + or - terms), each term can be differentiated separately and then added together afterwards to give the total differential.

Equation for x

First term: t2/2

Take the power which t is raised by (2) and multiply it by the coefficient of t(1/2), then drop the power by one.

d/dt [t2/2] = t

Second term: +1

The differential of a constant is always equal to 0.

d/dt [1] = 0.

... dx/dt = d/dt [t2/2 + 1] = d/dt [t2/2] + d/dt [1] = t + 0 =t

Equation for y

Repeat the process for y. This is a little tricker since the t is in the denominator of the first term. It is easier to perform the differential if we rewrite the term 4/t as 4t-1. The equation for y is now

y=4t-1-1

Using the same method as before, for the first term:

d/dt [4t-1] = -4t-2

and for the second term:

d/dt [-1] = 0

...dy/dt = d/dt [4t-1-1] = d/dt [4t-1] + d/dt [-1] = -4t-2 + 0 = -4t-2 = -4/t2.

Putting all this together

The equation we need to find the gradient is

dy/dx = dy/dt*dt/dx = dy/dt+1/(dx/dt)

We have already worked out dy/dt and dx/dt. To get dt/dx we just take the reciprocal of dx/dt (that is, switch the denominator and numerator round- in this case the denominator would be 1 as t=t/1).

dt/dx = 1/(dx/dt) = 1/t

Now dy/dx = dy/dt*dt/dx = -4/t2 * 1/t = (-4*1)/(t2*t) = -4/t3.

Now that we have an equation for the gardient, dy/dx, we can simply substiute in our value for t given in the question (t=2).

The gradient at t=2 is therefore:

dy/dx = -4/2= -4/(2*2*2) = -4/8 = -1/2

Abigail S. A Level Maths tutor, GCSE Maths tutor, A Level Physics tut...

9 months ago

Answered by Abigail, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

176 SUBJECT SPECIALISTS

£20 /hr

Sarah R.

Degree: Physics (Bachelors) - Exeter University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Economics
-Personal Statements-

“Hi! I'm Sarah and I'm a second year Physics student at the University of Exeter.  Maths and Physics have been a huge part of my education and general interests for a long time and I hope that I can pass on my knowledge and enthusiam t...”

£20 /hr

Angus S.

Degree: Cognitive Science (BSc Hons) (Bachelors) - Edinburgh University

Subjects offered: Maths

Maths

“Hello! My name is Angus and I'm 19 years old. Currently, I'm a first year Informatics student at the University of Edinburgh. I spent part of a year after school travelling and teaching English as a foreign language, although Maths wa...”

MyTutor guarantee

£24 /hr

Hinsum W.

Degree: Medicine (Bachelors) - Edinburgh University

Subjects offered: Maths, Chemistry+ 3 more

Maths
Chemistry
Biology
-Personal Statements-
-Medical School Preparation-

“Hi, I'm a first year medic at Edinburgh. I've had 7 years' worth of experience teaching children Maths and English and have mentored lower school peers, so I'm bound to have an approach that will suit your learning style!”

About the author

£20 /hr

Abigail S.

Degree: Clinical Science (Medical Physics) (Masters) - Kings, London University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Chemistry

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

For the curve f(x) = 2x^3 - 54x, find the stationary points and state the nature of these points

How to find the equation of a tangent to a curve at a specific point.

[FP2] Solve: 3 cosh x - 4 sinh x = 7

Find the integral between 4 and 1 of x^(3/2)-1 with respect to x

View A Level Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok