Mathematics 53 Midterm Exercise Set

I. Evaluate the following limits.

1. limx→2

x2 − 4

x2 − 5x+ 6

2. limx→1

x6 − 1

x8 − 1

3. limx→4

sin(x− 4)√x− 2

4. limx→0+

[[8− 10x

2

]]

5. limx→−∞

√3x2 + 2− 1

x+ 4

6. limx→+∞

3x√x2 − 1− x

7. f(x) =

2x+ 4, , x ≤ 5

2x2 − 1 , 5 < x < 7

x− 3 , x ≥ 7

Find: limx→7

f(x) and limx→5

f(x)

II. Continuity. Do as indicated.

1. f(x) =

sin(πx) , x ≤ −4√x+ 5− 2

x+ 1, x > −4

Discuss continuity of f . If discontinuous, identify the type of discontinuity.

2. g(x) =

k sin

((x+ 3)π

6

), x ≤ 2

3−√

11− xx− 2

, x > 2

Find the value of k such that g is continuous on [0, 11].

III. Finddy

dx. Do not simplify.

1. y =√

1 + sec (3x2 + 2x− 1) + csc (3x2 + 2x− 2)

2. y =(1 +√x)3 (

1− 2 3√x)4

3. y = (4x+ cos 3x)4 (4x2 − sinx)5

4. y =x− sin3 5x6√

1− cotx7

5. y =x3 − 4x2

tan (4x2 − π2)

6. 3 =√x2 − y cos y

7. x2 sin y − 4xy2 = x3 + 2

IV. Do as indicated.

1. Find the equation of the line tangent to the curve y = −5x2 + 3x and parallel to the line7x+ y − 5 = 0.

2. The function k(x) = x3 + 3x2 − 4 is continuous on [-2, 1] and differentiable on (-2, 1). De-termine the number c that will satisfy the conclusion of the Mean Value Theorem.

3. Give the linearization L(x) of y = cos(x2 + 1)− x at x = 0.

4. Given f(x) =4x2

x2 − 9, f ′(x) =

−72x

(x2 − 9)2, and f ′′(x) =

216(x2 + 3)

(x2 − 9)3.

(a) Find the domain and asymptotes of f .

(b) Accomplish the table determining the intervals for which f is increasing or decreasing,concave upward or downward, and all relative extrema and points of inflection of f .

(c) Sketch the graph of f .

V. Word Problems.

1. The position of a particle (with respect to the origin) moving along a horizontal line at tseconds is given by s(t) = 3t2 − 6t + 4. What is the total distance traveled by the particleafter 2 seconds?

2. A bacterial cell, spherical in shape, increased its radius from 2 µm to 2.5 µm. Approximatethe increase in the volume of the cell using differentials.

3. Water is running out of a conical funnel at the rate of 1000 mm3/s. If the radius of the baseof the funnel is 40 mm and the altitude is 80 mm, find the rate at which the water level isdropping when it is 20 mm from the top.

4. Find the dimensions of the largest rectangle that can be inscribed in a right triangle withlegs of length 3 inches and 4 inches if two sides of the rectangle lie along the legs.

5. A company estimates the total cost of producing x units of a certain item is given byC(x) = x4−222x3−900x2 +1500x pesos. Determine the production level that will minimizethe average cost.