By Frank Ayres; Elliot Mendelson

From http://www.atilim.edu.tr/~math153/000cal02.pdf

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**Extra info for Calculus Crash Course**

**Example text**

9 Let y = u3 and u = 4x2 - 2x + 5. Then the composite function y = (4x2 - 2x + 5)3 has the derivative: Note: In the second formulation of the chain rule, the y on the left denotes the composite function of x, whereas the y on the right denotes the original function of u (what we called the outer function before). 10 Differentiate y = (x2 + 4)2(2x3 - 1)3. Page 25 Higher Derivatives Let y = f(x) be a differentiable function of x, and let its derivative be called the first derivative of the function.

Also, . Because y" > 0 at x = 5, y has a minimum value (= 75) at x = 5. 4 Using 200 feet of wire, Alexandra would like to construct a rectangular garden consisting of three sides with the fourth side against a wall of the house. What are the dimensions of the garden that will yield the maximum possible area? Solution. We first begin by defining x = length of the garden side perpendicular to the house y = length of the garden side parallel to the house Given the total amount of fencing wire is 200 feet, then Also, the area of the rectangular garden is Solving Eq.

The equation of the parabola is y = (2/625)x2 and y'=4x/625. 8000 and φ = 380°40'. Hence the required angle is φ = 90° - θ = 51°20'. Figure 3-2 Page 39 Maximum and Minimum Values Increasing and Decreasing Functions A function f(x) is said to be increasing on an open interval if u < v implies f(u) < f(v) for all u and v in the interval. A function f(x) is said to be increasing at x = xo if f(x) is increasing on an open interval containing xo. Similarly, f(x) is decreasing on an open interval if u < v implies f(u) > f(v) for all u and v in the interval, and f(x) is decreasing at x = xo if f(x) is decreasing on an open interval containing xo.