# Calculus: An Integrated Approach to Functions and their by Robin J. Gottlieb By Robin J. Gottlieb

A big criticism of professors instructing calculus is that scholars do not have the ideal historical past to paintings during the calculus direction effectively. this article is focused without delay at this underprepared viewers. it is a single-variable (2-semester) calculus textual content that includes a conceptual re-introduction to key precalculus rules during the exposition as applicable. this can be the suitable source for these faculties facing poorly ready scholars or for faculties introducing a slower paced, built-in precalculus/calculus path.

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If an input of 8 produces an output of 3, then the point (8, 3) will be on the graph. More generally, if (x1, y1) is on the graph of f , then y1 = f (x1). The converse14 is also true. If y1 = f (x1), then the point (x1, y1) is on the graph of f . If a function is given by an equation y = f (x), then a point (x, y) lies on the graph of the function if and only if it satisfies the equation y = f (x). The “if and only if” construction comes up frequently in mathematical discussions, so we’ll pause for a moment to clarify the meaning of an “if and only if” statement.

A) Express the volume of the box in terms of s and h. (b) Express the surface area of the box in terms of s and h. (c) If the volume of the box is 120 cubic inches, express the surface area of the box as a function of s. 29. You are constructing a closed rectangular box with a square base and a volume of 200 cubic inches. If the material for the base and lid costs 10 cents per square inch and the material for the sides costs 7 cents per square inch, express the cost of material for the box as a function of s, the length of the side of the base.

Express r as a function of h. SOLUTION First, be sure you understand the question. The radius of the pudding skin and the radius of the bowl are generally different. They are the same only if the bowl is filled all the way to the top. 2 2" 2" 2-h r h h bowl of chocolate pudding 2 2 2-h r r h searching for a way to relate h and r... h success! 12 We can relate r and h by looking at a cross-sectional slice and using a right triangle. While it may be tempting to draw a triangle whose legs are h and r, this is not useful, since the hypotenuse of the triangle is unknown.