Rates of Change & the Derivative — Drills
Rates of Change & the Derivative has 6 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- Function — A rule pairing each input with exactly one output.
- Slope — How much y changes for each 1 that x increases.
- Average rate of change — f(b) - f(a)b - a over an interval.
- Tangent — tan θ = oppositeadjacent.
- Derivative — The slope of the tangent, written f'(x) or dydx.
- Difference quotient — f(x+h)-f(x)h, the slope over a gap of h.
- Quotient — The answer to a division.
Powers
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Differentiate f(x)=7x6.
Worked answer
42x^5
7 × 6 = 42
The power becomes 5
Problem 2
Differentiate f(x)=5x8.
Worked answer
40x^7
5 × 8 = 40
The power becomes 7
Average
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For f(x)=x2, find the average rate of change from x=3 to x=6.
Worked answer
9
36-96-3 = 273
= 9
Problem 2
For f(x)=x2, find the average rate of change from x=0 to x=6.
Worked answer
6
36-06-0 = 366
= 6
Evaluating
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y=3x3-6x, find the slope at x=4.
Worked answer
138
dydx = 9x2-6
9(16)-6 = 138
Problem 2
For y=4x3-4x, find the slope at x=0.
Worked answer
-4
dydx = 12x2-4
12(0)-4 = -4
Flat
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y=x2-12x+4, at what x is the tangent horizontal?
Worked answer
6
2x-12 = 0
x = 6
Problem 2
For y=x2-2x+1, at what x is the tangent horizontal?
Worked answer
1
2x-2 = 0
x = 1
Quotient
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Simplify the difference quotient (x+h)2-x2h.
Worked answer
2x+h
x2+2xh+h2-x2h
= 2xh+h2h = 2x+h
Constant
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Differentiate f(x)=17.
Worked answer
0
A constant never changes, so its rate of change is 0
Problem 2
Differentiate f(x)=23.
Worked answer
0
A constant never changes, so its rate of change is 0