UniGeniusAP Calculus AB and BC

T1.1 · Topic 1.1

Instantaneous change

Multiple choiceNo calculator

A ball's height is h(t)=20t−5t2h(t) = 20t - 5t^2 meters after tt seconds. For Δt≠0\Delta t \ne 0, the average velocity of the ball over the interval [1,1+Δt][1, 1 + \Delta t] is 10−5Δt10 - 5\Delta t meters per second. What is the instantaneous velocity of the ball at t=1t = 1?

Free response · 4 pointsNo calculator

Water flows into a tank. The volume of water in the tank at time tt minutes is V(t)V(t) liters. Selected values of V(t)V(t) are shown in the table.
Table of t in minutes and V(t) in liters: at t = 0, 2, 3, 3.5 and 4 the volume is 50, 58, 63, 65.8 and 69.
tt (minutes)V(t)V(t) (liters)
050
258
363
3.565.8
469
(a)
Find the average rate of change of VV over the interval 2≤t≤42 \le t \le 4. Indicate units of measure.
(2 points)

Write the setup as on paper, with units where asked.

(b)
Use the data in the table to give the best estimate of the instantaneous rate of change of VV at t=4t = 4. Explain why your choice of interval gives the best estimate the table allows.
(2 points)

Write the setup as on paper, with units where asked.

Multiple choiceNo calculator

The position of a cart is s(t)s(t) meters at time tt seconds. Selected values are shown in the table. What is the average velocity of the cart over the interval 1.5≤t≤2.51.5 \le t \le 2.5?
Table of t in seconds and s(t) in meters: t = 1, 1.5, 2, 2.5, 3 give s = 4, 5.25, 7, 9.25, 12.
tt (seconds)s(t)s(t) (meters)
14
1.55.25
27
2.59.25
312

Multiple choiceNo calculator

For h≠0h \ne 0, which expression gives the slope of the secant line to the graph of ff through the points where x=ax = a and x=a+hx = a + h?

Multiple choiceGraphing calculator

Let f(x)=2xf(x) = 2^x. With a calculator, evaluate the average rate of change of ff over [1,1+h][1, 1 + h] for h=0.1h = 0.1, 0.010.01 and 0.0010.001. To three decimal places, what value do these average rates approach as h→0h \to 0?

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