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Calculus Volume 1

Review Exercises

Calculus Volume 1Review Exercises

Review Exercises

True or False? Justify your answer with a proof or a counterexample. Assume that f(x)f(x) is continuous and differentiable unless stated otherwise.


If f(−1)=−6f(−1)=−6 and f(1)=2,f(1)=2, then there exists at least one point x[−1,1]x[−1,1] such that f(x)=4.f(x)=4.


If f(c)=0,f(c)=0, there is a maximum or minimum at x=c.x=c.


There is a function such that f(x)<0,f(x)>0,f(x)<0,f(x)>0, and f(x)<0.f(x)<0. (A graphical “proof” is acceptable for this answer.)


There is a function such that there is both an inflection point and a critical point for some value x=a.x=a.


Given the graph of f,f, determine where ff is increasing or decreasing.

The function increases to cross the x-axis at −2, reaches a maximum and then decreases through the origin, reaches a minimum and then increases to a maximum at 2, decreases to a minimum and then increases to pass through the x-axis at 4 and continues increasing.

The graph of ff is given below. Draw f.f.

The function decreases rapidly and reaches a local minimum at −2, then it increases to reach a local maximum at 0, at which point it decreases slowly at first, then stops decreasing near 1, then continues decreasing to reach a minimum at 3, and then increases rapidly.

Find the linear approximation L(x)L(x) to y=x2+tan(πx)y=x2+tan(πx) near x=14.x=14.


Find the differential of y=x25x6y=x25x6 and evaluate for x=2x=2 with dx=0.1.dx=0.1.

Find the critical points and the local and absolute extrema of the following functions on the given interval.


f(x)=x+sin2(x)f(x)=x+sin2(x) over [0,π][0,π]


f(x)=3x44x312x2+6f(x)=3x44x312x2+6 over [−3,3][−3,3]

Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down.


x ( t ) = 3 t 4 8 t 3 18 t 2 x ( t ) = 3 t 4 8 t 3 18 t 2


y = x + sin ( π x ) y = x + sin ( π x )


g ( x ) = x x g ( x ) = x x


f ( θ ) = sin ( 3 θ ) f ( θ ) = sin ( 3 θ )

Evaluate the following limits.


lim x 3 x x 2 + 1 x 4 1 lim x 3 x x 2 + 1 x 4 1


lim x cos ( 1 x ) lim x cos ( 1 x )


lim x 1 x 1 sin ( π x ) lim x 1 x 1 sin ( π x )


lim x ( 3 x ) 1 / x lim x ( 3 x ) 1 / x

Use Newton’s method to find the first two iterations, given the starting point.


y = x 3 + 1 , x 0 = 0.5 y = x 3 + 1 , x 0 = 0.5


y = 1 x + 1 = 1 2 , x 0 = 0 y = 1 x + 1 = 1 2 , x 0 = 0

Find the antiderivatives F(x)F(x) of the following functions.


g ( x ) = x 1 x 2 g ( x ) = x 1 x 2


f ( x ) = 2 x + 6 cos x , F ( π ) = π 2 + 2 f ( x ) = 2 x + 6 cos x , F ( π ) = π 2 + 2

Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes.


y = 1 x ( x + 1 ) 2 y = 1 x ( x + 1 ) 2


y = x 4 x 2 y = x 4 x 2


A car is being compacted into a rectangular solid. The volume is decreasing at a rate of 22 m3/sec. The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of 0.250.25 m/sec, find the rate at which the height is changing when the length and width are 22 m and the height is 1.51.5 m.


A rocket is launched into space; its kinetic energy is given by K(t)=(12)m(t)v(t)2,K(t)=(12)m(t)v(t)2, where KK is the kinetic energy in joules, mm is the mass of the rocket in kilograms, and vv is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of 1515 m/sec2 and the mass is decreasing at a rate of 1010 kg/sec because the fuel is being burned. At what rate is the rocket’s kinetic energy changing when the mass is 20002000 kg and the velocity is 50005000 m/sec? Give your answer in mega-Joules per second (MJ/s), which is equivalent to 106106 J/s.


The famous Regiomontanus’ problem for angle maximization was proposed during the 15th century. A painting hangs on a wall with the bottom of the painting a distance aa feet above eye level, and the top bb feet above eye level. What distance xx (in feet) from the wall should the viewer stand to maximize the angle subtended by the painting, θ?θ?

A point is marked eye level, and from this point a right triangle is made with adjacent side length x and opposite side length a, which is the length from the bottom of the picture to the level of the eye. A second right triangle is made from the point marked eye level, with the adjacent side being x and the other side being length b, which is the height of the picture. The angle between the two hypotenuses is marked θ.

An airline sells tickets from Tokyo to Detroit for $1200.$1200. There are 500500 seats available and a typical flight books 350350 seats. For every $10$10 decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard?

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