Taking and Estimating Derivatives
Rate of Change
Sin, Cos, Tan, Csc, Sec, Cot
Derivative Rules
Differentiation and Continuity
100
The derivative of a function f is given by f′(x)=0.2x+e0.15x. Which of the following procedures can be used to determine the value of x at which the line tangent to the graph of f has slope 2 ?
A - Evaluate 0.2x + e0.15x at x = 2.
B - Evaluate d/dx(0.2x+e0.15x) at x=2.
C - Solve 0.2x+e0.15x=2 for x.
D - Solve d/dx(0.2x+e0.15x)=2 for x.
C - Solve 0.2x+e0.15x=2 for x.
100
If f(x)=x6, then f′(x)=
f′(x)= 6x5
200
The graph of the function f, shown above, consists of three line segments. What is the average rate of change of f over the interval 1≤x≤7 ?
0
200
Which of the following statements, if true, can be used to conclude that f(3) exists?
- limx→3 f(x) exists.
- f is continuous at x=3.
- f is differentiable at x=3.
Statements 2 and 3 are true.
300
The function f is given by f(x)=1+2sinx. What is the average rate of change of f over the interval [0,π/2]?
4/π
400
The graph of the trigonometric function f is shown above for a≤x≤b. At which of the following points on the graph of f could the instantaneous rate of change of f equal the average rate of change of f on the interval [a,b] ?
At point B.
400
The graph of the function f, shown above, has a vertical tangent at x=3 and horizontal tangents at x=2 and x=4. Which of the following statements is false?
A - f is not differentiable at x=3 because the graph of f has a vertical tangent at x=3.
B - f is not differentiable at x=−2 and x=0 because f is not continuous at x=−2 and x=0.
C - f is not differentiable at x=−1 and x=1 because the graph of f has sharp corners at x=−1 and x=1.
D - f is not differentiable at x=2 and x=4 because the graph of ff has horizontal tangents at x=2 and x=4.
D - f is not differentiable at x=2 and x=4 because the graph of ff has horizontal tangents at x=2 and x=4.