Differentiation: Definition & Rules unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
Product rule
Maximum of f from a graph of f′
Derivative of eˣ and ln x
Recognizing a limit as a derivative
Cusp
f increasing ⟺ f′ positive
Derivative as slope of the tangent
Contrapositive as a working tool
Smooth join is the tangent line
Tangent vs normal line
Derivative of a piecewise function is piecewise
Differentiability implies continuity
Short answer 1. Define or explain: Estimate versus find
3 ptsShort answer 2. Define or explain: One-sided derivative
3 ptsShort answer 3. Define or explain: Table problems test the rules
3 ptsShort answer 4. Define or explain: Normal line
3 ptsFree response
9 ptsNO CALCULATOR. Two particles, H and J, are moving along the x-axis. For 0 ≤ t ≤ 5, the position of particle H at time t is given by x_H(t) = e^(t² − 4t), and the velocity of particle J at time t is given by v_J(t) = 2t(t² − 1)³.
A. Find the velocity of particle H at time t = 1. Show the work that leads to your answer.
B. During what open intervals of time t, for 0 < t < 5, are particles H and J moving in opposite directions? Give a reason for your answer.
C. It can be shown that v′_J(2) > 0. Is the speed of particle J increasing, decreasing, or neither at time t = 2? Give a reason for your answer.
D. Particle J is at position x = 7 at time t = 0. Find the position of particle J at time t = 2. Show the work that leads to your answer.