Rates of Change & the Derivative — Practice
Average speed over a shorter and shorter trip settles on the speed at one instant.
What these questions 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.
Problem 1
For f(x)=x2, find the average rate of change from x=1 to x=4.
Worked answer
5
16-14-1 = 5
Problem 2
For f(x)=3x+2, find the average rate of change from x=0 to x=5.
Worked answer
3
17-25 = 3
Problem 3
Differentiate f(x)=x4.
Worked answer
4x^3
4x3
Problem 4
Differentiate f(x)=x7.
Worked answer
7x^6
7x6
Problem 5
Differentiate f(x)=9.
Worked answer
0
A constant never changes, so the rate is 0
Problem 6
Differentiate f(x)=3x2.
Worked answer
6x
3 × 2x = 6x
Problem 7
Differentiate f(x)=x3-6x.
Worked answer
3x^2-6
3x2 from the first, -6 from the second
Problem 8
Differentiate f(x)=5x2+2x-7.
Worked answer
10x+2
10x + 2 + 0
Problem 9
For y=x3-6x, find the slope at x=2.
Worked answer
6
dydx = 3x2-6
3(4)-6 = 6
Problem 10
For y=x2, find the slope at x=-3.
Worked answer
-6
dydx = 2x
2(-3) = -6
Problem 11
Simplify the difference quotient (x+h)2-x2h.
Worked answer
2x+h
2xh+h2h
= 2x+h
Problem 12
Using the difference quotient, find f'(x) for f(x)=x2.
Worked answer
2x
The quotient simplifies to 2x+h
As h → 0 that becomes 2x
Problem 13
For y=x2-8x+1, at what x is the tangent horizontal?
Worked answer
4
2x-8 = 0
x = 4
Problem 14
For y=x3-3x, find the positive x where the tangent is horizontal.
Worked answer
1
3x2-3 = 0
x2 = 1, so x = 1 or -1
Problem 15
At a point the derivative is negative. What is the function doing?
- Falling
- Rising
- Flat
- Undefined
Worked answer
Falling
A negative slope means the curve is going down
Problem 16
For y=4x3, find the slope at x=1.
Worked answer
12
dydx = 12x2
12(1) = 12