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Jump to Page. Search inside document. In easier groups of exercises, representative solutions are shown. When appropriate, only the answer is listed. When possible, we tried to make each piece of art with the same scale to show a realistic and consistent graph. Note: Notes to the instructor pertaining to hints on instruction or conventions to follow.
Wz 65,then5 —2 0. Equating the two surface areas yields 2nr? See Figure Since the domain of fis R, any value of 9 z is in its domain.
Thus, the domain is —c0, 1] U [2, Since f 2 is always in the domain of g, the domain of 9 fis the same as the domain of f, 15, This —5, This is always true since the result of a square root is nonnegative.
The domain is —co, 3]. Hence, the domain of f 0 g is -5, —4] U [4, 5]. Note: In anticipation of using the chain rule, you may want to think of the last Domain of g calculator operation if you were evaluating on a calculator as your choice for y.
Square both sides to obtain an equation in either sin or cos. Since each side of the equation was squared, the solutions must be checked in the original equation, x is an extraneous solution. BD tim. Round-off will affect answers. The values in the tables were found using double precision.
Since we cannot enter arbitrarily amall values on a calculator, we cannot even begin to use a calculator to prove that a limit exists. Hence, 5. Thus, if we choose 6 less than or equal to the minimum of —Lo?
If ze fa? These lines intersect the graph of fat points with coordinates Ya? This is a contradiction and the limit does not exist. Figure 83 Figure 54 jimilar to that of Exercise The right-hand limit is necessary because the object must be between the focus and the lens.
Thus, LS G2 lim, cos DNE since cos does not approach a real number L, but rather oscillates between —1 and 1 as z increases without bound. The horizontal asymptote is found by finding Jim f z and J 2.
Since each additional gallon of water contains 0. Hence, fis continuous on 0, This line has zintercept 3. No creature is hit. Hence, the velocity is positive on the following intervals: 0, 0. Since fis always positive, no cusp is formed. The domain of f' is R. OOOO ui 2 6. If this line has z-intercept 4, then 4, 0 must satisfy its equation. Wweekye Phas! The volume of the gas is decreasing because the pressure is increasing. Thepoint is f, At Figure 39 Figure 40, [HU] The 2-coordinates of points at which the graph has a horizontal tangent line are 0.
Gas He 2 21h cosPer — sn? Note: For all exercises, we assume is that the denominators are nonzero. Note: In Exercises , the first equation shown is the result of differentiating the original equation implicitly.
Say — 3y? BD 22? Figure 35 G2? In all cases, they are perpendicular. AI 32? Thus, Go-aa 30 1. The volume of the ice outer hemisphere — inner hemi. BB Orient the plane as in Figure L is the distance CNL position of plane between the plane and the control tower.
Vix dwh o? Weep g ae — ages gine Wo share h RE w —0. Assuming no slippage in the chain, the length of chain moving around each sprocket must be equal. The length 2 that the wheel travels ia 14g. Using similar triangles and. Figure 71  Let z and y denote the distances from the intersection of A and B, respectively, and J the distance between A and B, [? Distance is decreasing. Thus, A is located at 2cos4nt, 2sin4ni.
Bis moving to the left. Also, cos! Ifa 0, then f' a 0. Hence 0 is the only critical number of f. This does not contradict Theorem 4. Hence there are no critical numbers. Hence, by 3. For example, if a graph is marked in units of 0. MIN: 0. After drawing a line between the two endpoints of the graph, a is to find the values at which the slope of the tangent line is equal to the it appears that these values are approximately 3 and 7.
Note: In Exercises , unless otherwise specified, the functions are continuous on the indicated closed interval and differentiable on the associated open interval, thereby " satisfying the hypotheses of the mean value theorem MVT.
This is not a contradiction, because fis not continuous on the interval —1, 4. More From aleven quispe. Alan Sebastian Aguero. Christian Valerio. Braian Nahuel.
CÁLCULO con geometría analítica
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