Calculus with Analytic Geometry I–Fall 2013 Sample Final Exam Don’t panic! This is MUCH longer than the actual exam will be, and probably harder. More than a sample exam it is a study guide. We’ll discuss these problems next week (12/2, 12/4), but it would be so much more useful, and helpful, and wonderful, if you actually had worked on them. So will you, pretty please, at least look at them. 1. Compute the following limits. Use any procedure you want, but show ALL work. x2 − 7x + 10 . x→2 x2 − 4 p (b) lim x2 − 3x + 5 − x . (a) lim x→∞ (c) limx→∞ x100 e−3x . 3x 2 (d) lim 1 + . x→∞ x (e) lim ln x. x→0+ (f) lim x0.001 ln x. x→0+ 2. 3. Evaluate the following derivatives. Since the assumption is that you did lot of exercises and now are at least as good as any calculator in finding derivatives, it is OK to just write the answer. (a) y = (sin 4x)(cos2 3x), find √ dy . dx ln x + , find f 0 (x). x q (c) g(x) = sin( ex2 + sinh(x)). (b) f (x) = ex x 4. Find the tangent line to the curve of equation y 2 − xy − 5x3 = 1 at the point in the first quadrant where x = 1. 5. For the following function, in the given domains: (a) Determine the intervals of increase and of decrease. (b) Find all critical points and classify them as relative maximum, minimum, or neither. (c) Find the absolute maximum value, or decide there is none. Same for the absolute minimum. (d) Determine the intervals of concavity. (e) Find all inflection points. (f) find all horizontal and vertical asymptotes. (g) Use the preceding information to sketch a graph of the function on the grid provided on the last page of this exam. Draw the axes and indicate the unit on each axis. (a) f (x) = x5 − 2x4 , −∞ < x < ∞. (b) g(x) = x2 e−x , 0 ≤ x < ∞. 2 (c) h(x) = e−x , −∞ < x < ∞. x − x2 , −∞ < x < ∞. x2 + 1 2 (e) G(x) = x + , −∞ < x < ∞. x (d) F (x) = 6. The length of a rectangle is increasing at the rate of 6cm/sec, and its width is increasing at the rate of 7cm/sec. When the length is 5 cm and the width is 8 cm, how fast is the area of the rectangle increasing? 2 7. Find the maximum and the minimum values of the following functions in the given intervals. If there is none, say so. In all cases, justify your answer. (a) f (x) = x3 − 6x2 + 9x + 5, 2 ≤ x ≤ 4. (b) f (x) = x| ln x|, 0 < x ≤ 1. x−1 (c) f (x) = 2 , 0 ≤ x < ∞. x +1 8. (Almost the same as 4.7, # 49) An oil refinery is located on the north bank of a straight river that flows east to west. The river is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the the refinery. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $750,000/km under the river to the tanks. To minimize the cost of the pipeline, where should P be located? 9. Textbook, Section 4.7, Exercise # 64. 10. Textbook, Section 4.7, Exercise # 70. 11. Textbook, Section 4.7, Exercise # 71. 12. What are the dimensions of the largest rectangle with its lower-left vertex at (0, 0) and its upper=right vertex on the curve y = e−x . The figure below shows one such rectangle. 13. An object moves along a line. If its acceleration is given by a(t) = −2t (in ft/s2 , t being measured in seconds)), and its initial velocity is 9 ft/s. (a) Find its velocity function v(t) (b) For the time interval 0 ≤ t ≤ 5, find the total distance travelled. (Remember, that total distance is the amount that would be registered on an odometer. It is not necessarily displacement.) 14. Of the following formulas, some are true, some false. Decide which one is true and which one is false, justify your answer! Z (a) 2 2 ex dx = ex + C. and 3 Z 3 sin4 x dx = −1/4 (sin (x)) cos (x) − 3/8 cos (x) sin (x) + 3/8 x + C. Z 1 x 1 dx = + arctan x + C. (c) (1 + x2 )2 2 1 + x2 Z x (d) sin t dt = cos t + C. (b) 0 Z 3 (e) −x2 Z dx = 1 3 2 e−t dt. 1 15. Compute the following integrals. 2 Z 2 √ 1 2 3 (a) x x+ √ dx. 3 x 0 Z 3 2x3 (b) dx. 4 0 x +5 Z ex dx. (c) 1 + e2x Z (d) (x4 + 1)6 x3 dx. 16. Find the derivative of the following functions Z sin x t2 (a) g(x) = dt. 1 + t4 1 Z 3x2 t e (b) h(x) = √ dt. t x 17. Use the picture to show that Z 1 Z arctan x dx + 0 π/4 tan x dx = 0 18. Show that the equation 2x − cos x = 0 has exactly one real root. π . 4 4 1 19. Calculate the area of the shaded region. The curve shown has equation y = 1 + x3 . 2 Z 20. Let f (x) = 0 x 1 − t2 dt. Answer (or do) the following: 3 + t4 (a) In what intervals is f increasing? (b) In what intervals is f decreasing? (c) In what intervals is f concave up? (d) In what intervals is f concave down? (e) List all critical points of f and classify them as local maximum, minimum, neither. (f) List the x coordinate of all inflection points.