Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 5. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Representations of Functions 21 ii. Using the information from part i we label the point P on the graph 63 7 . iii. The volume of liquid in this beaker V is 9nh. We want to calculate height given volume so we solve for h in terms of V V V h so Cb V w w The height is proportional to the volume. We can see this from the function formula h Cft V 9 V or from the physical situation itself. The walls of the beaker are perpendicular to the base therefore all cross-sections are equal in area and the height is proportional to the volume. In the examples that follow we will work on expressing one variable as a function of another. These examples are chosen to highlight relationships that will arise repeatedly throughout our studies. They will also serve as a review of some geometry including similar triangles and the Pythagorean Theorem. For a summary of some useful geometric formulas refer to Appendix B. REMARK Examples in a mathematics text are meant to be read actively with a pencil and paper. A solution will have more impact and stay with you longer if you have spent a bit of time tackling the problem yourself. Read the problems that follow and try each one on your own before reading the solutions. The problem-solving strategies highlighted below should help you out. Think of them as a way of coaching yourself through a problem. Portable Strategies for Problem Solving Draw a picture whenever possible. Label known quantities and unknown quantities so you can refer to them. Make your labeling clear and explicit. Take the problem apart into a series of simpler questions. Plan a strategic approach to the problem. Make the problem more concrete or simpler in order to get started. Either solve the problem with the concrete numbers and try to generalize or spot check your answer by making sure it works in a concrete example or two. EXAMPLE Functioning around the house. An 8-foot ladder is leaning against the wall of a house. If the foot of the ladder is x feet from the
