Question: Determine whether the statement is true or
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
1. If f'(c) = 0, then f has a local maximum or minimum at c.
2. If f has an absolute minimum value at c, then f'(c) = 0.
3. If f is continuous on (a, b), then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c and d in (a, b).
4. If f is differentiable and f (-1) = f (1), then there is a number c such that |c| 5. If f'(x) 6. If f"(2) = 0, then (2, f (2)) is an inflection point of the curve y = f (x).
7. If f'(x) = g'(x) for 0 8. There exists a function f such that f (1) = -2, f (3) = 0, and f'(x) > 1 for all x.
9. There exists a function f such that f (x) > 0, f'(x) 0 for all x.
10. There exists a function f such that f (x) 0 for all x.
11. If f and g are increasing on an interval I, then f + g is increasing on I.
12. If f and g are increasing on an interval I, then f – g is increasing on I.
13. If f and g are increasing on an interval I, then fg is increasing on I.
14. If f and g are positive increasing functions on an interval I, then fg is increasing on I.
15. If f is increasing and f (x) > 0 on I, then g (x) = 1/f (x) is decreasing on I.
16. If f is even, then f' is even.
17. If f is periodic, then f' is periodic.
18. The most general antiderivative of f (x) = x-2 is F (x) = -1/x + C.
19. If f'(x) exists and is nonzero for all x, then f (1) ≠f (0).
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> A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450m above the ground. (a). Find the distance of the stone above ground level at time t. (b). How long does it take the stone to reach the ground? (c). With what velocit
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> Evaluate the integral. f10 eπt dt
> Evaluate the integral. f10 sin (3πt) dt
> Evaluate the integral. f10v2 cos (v3) dv
> Evaluate the integral. f csc2x/ 1 + cot x, dx
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> Evaluate the integral. f10 (4√u + 1)2 du
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> Evaluate the integral. f10 (1 – x)9 dx
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> Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents. y= f(x) 6 2. 2.
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> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> A light is to be placed atop a pole of height feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle (see the figure) and inv
> A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?
> Sketch the graph of a continuous, even function f such that f (0) =, f'(x) = 2x if 0 < x < 1, f'(x) = -1 if 1 < x < 3, and f'(x) = 1 if x > 3.
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> Show that if f is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of fba f (x) dx.
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 0, x= 9 – y?; about x = -1
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> Find f (x). f"(x) = 2x3 + 3x2 - 4x + 5 f (0) = 2, f(1) = 0
> Find f (x). f"(x) = 1 - 6x + 48x2 f (0) = 1 f'(0) = 2
> Find f (x). f' (u) = u2 + √u /u, f (1) = 3
> Find f (x). f'(t) = 2t - 3 sin t, f (0) = 5
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> Find the local and absolute extreme values of the function on the given interval. f(x) = x + sin 2x, [0, 7]
> 1. If f and g are continuous on [a, b], then 2. If f and g are continuous on [a, b], then 3. If f is continuous on [a, b], then 4. If f is continuous on [a, b], then 5. If f is continuous on [a, b] and f (x) > 0, then 7. If f and g are cont
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> The velocity of a wave of length L in deep water is where K and C are known positive constants. What is the length of the wave that gives the minimum velocity? L v = K- C C L
> An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sig
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> Find the area of the region bounded by the given curves. y = x², y= 4x – x²
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> If you have a graphing calculator or computer, why do you need calculus to graph a function?
> Suppose that you push a book across a 6-meter-long table by exerting a force f (x) at each point from x = 0 to x = 6. What does f60f (x) dx represent? If is measured in newtons, what are the units for the integral?
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