CALC PICK 10
16. let f be the function given by f(x) = x^3 - 6x^2 - 15x what is the max value of f on the interval [0,6] A. 0 B. 5 C. 6 D. 8
answer: A why/process: critical points / find derivative & set = to 0 and solve to get x= 5 (-1 is not on the interval) then make a sign chart and whatever goes from + to - is a max
9. when x = 2e, lim h--> 0 in(x+h) - ln (x) / h is A. 1/2e B . 1 C. ln 2e D. Nonexistent
answer: A why/process: derivative in disguise ln (x) = 1/x so ln (2e) = 1/2e
1. if f is the function given by f(x) = 4/x + 5x - 1 then f'(2) = A. 4 B. 6 C. 7 D. 11
answer: A why/process: find derivative and plug in 2
24. let f be the function given by f(x) = 2x^2 + 14x - 16 / x^2 - 9x + 8 for what values of x does f have a removable discontinuity A. 1 only B. 8 only C. -8 and 1 D. 1 and 8
answer: A why/process: take out 2 from the top and factor top (x-1) cancels out making it a discontinuity at 1
80. the first derivative of the function f is defined as f'(x) = (x^2 +1) sin (3x - 1) for -1.5 < x < 1.5 on which of the following intervals is the graph of f concave up? A. B. C. D.
answer: A (-1.5, -1.341) and (-0.240, 0.964) why/process: plug eq into y1 and nderiv into y2 and graph above on graph = concave up
4. if f'(x) = 3x^2 + 2x and f(2) = 3 then f(1) = ? A. -10 B. -7 C. 10 D. 13
answer: B why/process: integrate f'(x) then plug in f(2) and find +c to = 3 then plug in 1
5. (table) during an evacuation drill people leave a building at a rate of R(t).. using a right reimann sum with 3 sub intervals A. 230 B. 1150 C. 1375 D. 2075
answer: B why/process: A = 5(100+75+55)
19. lim x--> 0 4x^2/e^4x - 4x - 1 is A. 0 B. 1/2 C. 8 D. Nonexistent
answer: B why/process: LH rule derive top and bottom separately derive again to get 1/2e^4x = 1/2
20. let g be a twice diff increasing function of t. if g(0) = 20 and g(10) = 220 which of the following must be true on the interval 0 < 1<10? A. g'(t) = 0 for some t in the interval B. g'(t) =20 for some t in the interval C. g''(t) = 0 for some t in the interval D. g''(t) > 0 for some t in the interval
answer: B why/process: MVT y2 - y1 / x2 - x1
10. if dy/dx = x^4 - 2x^3 + 3x - 1 then d^3y/dx^3 evaluated at x = 2 is A. 11 B. 24 C. 26 D. 125
answer: B why/process: derive once for y'' then again for y''' then plug in 2 to y'''
12. given that 3x - tan y = 4 what is dy/dx in terms of y? A. 3sin^2 y B. 3cos^2 y C. 3cosy cot y D. 3/1+9y^2
answer: B why/process: drive and get y' by itself to get 3/sec^2 y 1/Sec^2 = cos^2 so bring it to the top
76. the graph of f' is shown above for -3<x<3 on what intervals is f increasing A. [-3,-1] only B. [-1,3] only C. [-2,0] and [2,3] D. [-3, -1] and [1,3]
answer: B why: above in f' = positive slope on f
82. let f be a function such that f(1)= -2 and f(5) = 7 which of the following conditions ensures that f(c) = 0 for some values c on the open interval (1,5) A. B. C. D.
answer: B f is increasing on the closed interval of [1,5] intermediate value theorem
13. for time t > 1 the position of a particle moving along the x-axis is given by p(t) = √t - 2. tag what time t in the interval 1<t<16 is the instantaneous velocity = to the average velocity A. 1 B. 121/25 C. 25/4 D. 25
answer: C why/process: 1/2√t = p(16) - p(1) / 16 - 1 1/2√t = 1/5 solve to = 25/4
90. selected values of the increasing function h and its derivative h' are shown in the table above. if g is a differentiable function such that h(g(x)) = x for all x what is the value of g'(7) ? A. -1/10 B. 1/10 C. 1/5 D. 7/5
answer: C g'(7) = 1/h'(3) = 1/5
28. lim x--> infinity 3+2^x/ 4-5^x is A. -2/5 B. 0 C. 3/4 D. Nonexistent
answer: C why/process" top heavy/ bottom heavy & simply to 3/4
6. if y = x^2 (e^x - 1) then dy/dx = A. 2xe^x B. 2xe^x - 2x C. x^2e^x + 2xe^x - 2x D. x^2e^x + 2xe^x - x^2 - 2x
answer: C why/process: product rule f'g+g'f
18. let f be the function defined by f(x) = ∛x what is the approximation for f(10) found by using the line tangent to the graph of f at the point (8,2) A. 11/6 B. 25/12 C. 13/6 D. 7/3
answer: C why/process: y-y1 = m (x-x1) find derivative and plug in 8 to get m plug in x,y and m to equation to get answer
79. the graph of the function f is shown above. of the following intervals, on which is f continuous but not differentiable A. (0,1) B. (1,2) C. (2,3) D. (3,4)
answer: C why: cusp
23. using the substitution u = x+ 1 ∫x/√x+1 is equivalent to A. B. C. D.
answer: C ∫u ^1/2 - u ^-1/2) du why/process: du=dx ∫u-1/u^1/2
3. the graph of y = f(x) is shown above. what is lim x --> 1 f(x) A. 0 B. 1 C. 3 D. the limit does not exist
answer: D Why: jump discontinuity
8. the graph of f'', the second derivative of the function f is shown above. which could be the graph of f A. B. C. D.
answer: D why/process: above on f'' = concave up of f below on f'' = concave down on f
27. the positive variables of p and c change with respect to time t. the relationship between p and c is given by the equation p^2 = (20-c)^3. at the instant when dp/dt = 41 and c = 15 what is the value of dc/dt? A. -82/75 B. -2√5/3 C. -3√5/2 D. -82√5/15
answer: D why/process: solve for p using equation (plug in 15) 2p dp/dt = 3(20 - c) ^2 (-1) dc/dt plug in all we know and solve for dc/dt
14. if f is a differentiable function and y = sin (f(x^2)) what is dy/dx when x = 3 A. cos (f'(9)) B. 6 cos (f'(9)) C.f'(9) cos (f(9)) D. 6f'(9) cos (f(9))
answer: D why/process: y' = cos (f(x^2)) (f'(x^2)) (2x) then plug in 3 for x
77. the rate at which water leaks from a tank in gallons per hour is modeled by R a differentiable function of the number of hours after the leak is discovered. which od the following is the best interpretation of R'(3) A. B. C. D.
answer: D why: R' would be per hour per hour
26. the table above gives values if the continuous function f at selected values of x. if f has exactly 2 critical points on the open interval (10,14) which of the following must be true A. f(x) > 0 B. f'(x) exists C. f(x) < 0 D. f'(12) ≠ 0
answer: D why: would change direction too many times
