Questions in Section Calculus I: page 6

Let f(x)=(x-2)^2(3-x). Without expanding the brackets, using methods of calculus, sketch the graph of the function showing all intercepts and exact x-values of the turning points. Find the closed area bounded by the curve and the x-axis in the first quadrant.

Let f(x)=(x-2)/(x^(2)-1).

Evaluate appropriate limits or derivatives to sketch y= f(x), showing all critical points. Clearly state the domain and range of f, and whether or not (with reason) the function is invertible.

Determine the value of the following limit: lim_(x->-oo)sqrt(5x^3-x)/(x^5+5).

Suppose lim_(x->a)f(x)=3 and lim_(x->a)g(x)=2. What is lim_(x->a)(f(x)-g(x)-1)/(2f(x)+g(x)-8)?

Suppose f(x) and g(x) are continuous functions on the interval [0,1] such that f(0)<g(0) and f(1) >g(1):

1. Find and sketch two functions which meet given criteria. Show that they intersect somewhere in the interval [0,1].
2. Use the intermediate value theorem and the function h(x)=f(x)-g(x) to show there must be a "c" such that 0<c<1 and f(c) = g(c).
3. If f(x) and g(x) are not continuous, is part (b) still true? If so, explain. If not, provide a counterexample.

Show that lim_(x->0)(x^2+1)sin(x)/x=1 in the two following ways:

1. using the squeeze theorem;
2. using the properties of limits.

The cost function for production of a commodity is c(x)=339+25x-0.09x^2+0.0004x^3.

1. Find the marginal cost function.
2. Find c'(100). What does it predict?
3. Find the error in the prediction.

x^y=y^x. Find (dy)/(dx).

Find first 2 non-zero terms of the Taylor series for the function y=xsinx near the point x=0.

Find first 3 non-zero terms of the Taylor series for the function y=xe^x near the point x=0.