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`.

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