A Preview of Calculus

Two ancient questions

The tangent problem and velocity

The area problem

Why limits unify the course

What exactly does the tangent problem ask the calculus student to find?

For a position curve s(t), the average velocity between t = a and t = b is (s(b) - s(a))/(b - a). What goes wrong if we set b = a to get the velocity at the instant a?

In the rectangle approximation of the area under a curve, what happens as the strips are made thinner and more numerous?

Which single concept do both the derivative and the definite integral reduce to?

A car's speedometer reports the number that calculus calls instantaneous velocity. Which construction produces that number from the position function?

Which founding problem of calculus corresponds to computing the area of a region bounded by a curve?

A triangle of base 4 and height 8 has area one half base times height. What is its area? Answer with a number.

The position is s(t) = 5t. What is the average velocity, the secant slope, between t = 1 and t = 3? Answer with a number.

Setting b equal to a in the difference quotient produces the meaningless fraction 0 over 0. What two-letter name is given to this form? Answer in lowercase.

The preview names two founding problems of calculus, the tangent problem and the area problem. How many founding problems are there? Answer with a number.