This is intended to be revision. Same rules as before. Do the work I set and you will get an A grade. Not enough of you are doing so, already. You need to be thoroughly up to speed on these two topics before we start the second half of term. If you have to choose, make sure the principles of differentiation are secure before revising co-ordinate geometry. Both will be used (not revised) in the second half of term.
1 Write the equation, preferably in the ax+by=c form, of the tangent to
y = 3x² – 11x + 4 that is parallel to y = x.[6]
2 Find the equation of tangent to y = 7x² that is parallel to y = 2x + 1.[5]
Find the tangent perpendicular to this, and [4]
show that these two tangents meet at (3/56, -3/40).[4]
3 Write the condition for (x,y) to be the same distance from (h,k) as from the line y=c.
Rewrite this on the assumption that (h , k) is the origin. Classify (identify) the curve.[3,3]
4 If x = 2at and t is a variable (possibly time), what is the value of dx/dt? [2]
If y = 20-5t², state dy/dt. When is y a maximum? Show that you have a maximum.[4]
Write dy/dx, and use this to describe the sort of curve that would be sketched on x,y axes.[4]
Differentiation Practice
Do the lot: Straight, Chain Rule, Product rule and nasties. Written in sets of four.
5 d (4x³ - 5x-⅔)
dx
6 d (5t⁻¹ + 2t1/5)
dt
7 d (⁴√t³ – 6/√t)
dy
8 d (4z+5)²
dz
9 d (p² + 5)³
dp
10 d (4x³ - 3)⁻¹
dx
11 d (1 / (1-2t))³
dt
12 d (sin x)⁴
dx
13 d (3 + x) (4 - x²)
dx
14 d (4x - 5) (3 + x²)
dx
15 d (3 + y)² (4 – y)
dy
16 d (2+z)
dz (3- z)
17 d t³
dx
18 d (xt)
dx
19 d (x + xy + yt)
dt
20 d (ø sin² ø)
dø
(sin x)⁴ is exactly the same as, and is usually written as, sin⁴ x, but sin⁻¹ x means the inverse function invsin x or arcsin x.
For the obtuse. And those having a bad day:
1 the differentials will have the same gradient, one.
6 (2t1.2 – 25)/5t²
11 6/(1-2t)⁴
15 (3+y)(5-3y)
19 y + (x+t) dy/dt + (1+ y) dx/dt
