Gateway 4 Practice Test

NO CALCULATORS MAY BE USED ON THIS TEST. SHOW YOUR WORK.

1.

(20 pts) Find the absolute minimum and maximum of f(x) = 15x⁴-15x²+31 on the interval [-1,2].

2.

(10 pts) Find the points where f has a local maximum or minimum on the given domain and identify each point as a local maximum or local minimum.

$$f(x) = x^2 + \frac{3}{x}, \;\; 0 < x < \infty.$$

3.

(28 pts) For the given derivative of a function f, f'(x) = (x+1)(x+2),

(a)

What are the critical numbers of f?

(b)

On what intervals is f increasing?

(c)

On what intervals is f decreasing?

(d)

At what points, if any, does f assume a local maximum or local minimum value?

4.

(12 pts) The graphs of the first and second derivative of a function y = f(x) are shown. Add to the picture a sketch of the approximate graph of f, given that the graph passes through the point P.

5.

(20 pts) The accompanying figure shows a portion of the graph of a twice-differentiable function y = f(x). At each of the five labeled points, classify y' and y'' as positive, negative or zero.

6.

(10 pts) Sketch the graph of a function that satisfies the given conditions. No formulas are required - just label the coordinate axes and sketch an appropriate graph.

$$f(-2) = -1,\;\;f(0) = 0,\;\;f(2) = 1,\;\;\lim_{x\rightarrow -\infty}f(x) = 0,\;\; \lim_{x\rightarrow \infty}f(x) = 0$$