91Ó°ÊÓ

State the Fundamental Theorem of Calculus.

Determine which value best approximates the definite integral. Make your selection on the basis of a sketch. \(\int_{0}^{1 / 2} 4 \cos \pi x d x\) (a) 4 (b) \(\frac{4}{3}\) (c) 16 (d) \(2 \pi\) (e) \(-6\)

An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by \(d h / d t=1.5 t+5\) where \(t\) is the time in years and \(h\) is the height in centimeters. The seedlings are 12 centimeters tall when planted \((t=0)\). (a) Find the height after \(t\) years. (b) How tall are the shrubs when they are sold?

The rate of growth \(d P / d t\) of a population of bacteria is proportional to the square root of \(t\), where \(P\) is the population size and \(t\) is the time in days \((0 \leq t \leq 10)\). That is, \(d P / d t=k \sqrt{t}\). The initial size of the population is \(500 .\) After 1 day the population has grown to 600 . Estimate the population after 7 days.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the norm of a partition approaches zero, then the number of subintervals approaches infinity.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is increasing on \([a, b]\), then the minimum value of \(f(x)\) on \([a, b]\) is \(f(a)\)

A ball is thrown vertically upward from a height of 6 feet with an initial velocity of 60 feet per second. How high will the ball go?

The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time \(t\) in seconds. How long will it take the rock to hit the canyon floor?

A baseball is thrown upward from a height of 2 meters with an initial velocity of 10 meters per second. Determine its maximum height.

Lunar Gravity On the moon, the acceleration due to gravity is \(-1.6\) meters per second per second. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?