MATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALS PDF

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MATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALSMATH 151 FINAL PRACTICE FINAL REVIEW MATERIALS...

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BC0F366F-C377-4413-85CF-3DA950FFE5EB final-exam-707d8

MATH 150 D100 / MATH 151 D100

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1. Evaluate the following limits. [3]

(a)

lim √

t→∞

t t2 + 1

)2x ( 1 lim 1 + x→∞ x

[3]

(b)

[4]

(c) Use the definition of the derivative to find the derivative of f (x) = l’Hospital’s rule.

√ x. Do NOT use

32668C5D-59EC-48CE-9401-B9D26AF64D7F final-exam-707d8

MATH 150 D100 / MATH 151 D100

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2. All parts of this question involve finding derivatives. You do not need to simplify your answers. 2

[3]

ex +1 , find f ′ (x). (a) If f (x) = x sin(x)

[4]

(b) If ln(y) + x = x2 + x cos(y), find

dy in terms of x and y. dx

CD9CCE2B-7247-4415-AF90-1FCBBB6A763D final-exam-707d8

MATH 150 D100 / MATH 151 D100

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3. Suppose f (x) =

x2 + 3 , x−1

f ′ (x) =

(x + 1)(x − 3) , (x − 1)2

f ′′(x) =

8 . (x − 1)3

[3]

(a) Determine the critical points of f .

[3]

(b) Determine the intervals on which the function f is increasing, and those on which the function f is decreasing. Classify the critical points as either local maxima, local minima, or neither.

[3]

(c) Determine where f is concave up and where it is concave down. Identify any inflection points.

(This question is continued on next page)

8CB66EA8-15D4-41A4-9AD3-A4C02BFC01C7 final-exam-707d8

MATH 150 D100 / MATH 151 D100

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4. This is a continuation of question 3. [3]

(a) Does f have any horizontal, vertical or slant asymptote? If so, find them.

[2]

(b) Indicate which of the following graphs is the graph of y = f (x) by circling the letter below the graph of your choice.

y

y

3 1 1

x

x

−1

(A)

(B)

y

y

3 1 1

(C)

x

3

(D)

x

674AABE9-C3D5-46B1-B7C7-8D9C6A7300D6 final-exam-707d8

MATH 150 D100 / MATH 151 D100

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5. Suppose that the only information we have about a function f is that f (1) = 3 and the graph of its derivative is as shown. y

y = f ′ (x) 1 x 0

1

[4]

(a) Use a linear approximation to estimate f (0.9) and f (1.1).

[3]

(b) Are your estimate in part (a) too large or too small. Explain.

178F73EB-883D-4CB4-B0ED-CDF203929B1E final-exam-707d8

MATH 150 D100 / MATH 151 D100 [10]

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6. Theorems and Definitions: Match the start of each theorem/definition with is conclusion. Write the corresponding letter in the space provided next to the conclusion. Be aware there are some statement beginnings that do not have a matching conclusion. (a) The Mean Value Theorem states that . . . (b) The Squeeze Theorem states that . . . (c) A critical number is a number that . . . (d) A function f is continuous at a number a . . . (e) The Extreme Value Theorem states that . . . (f) The natural number e is . . . (g) The graph of a function on an interval I is increasing if . . . (h) An inflection point is a point . . . (i) The derivative of a function f at a number a is . . . (j) L’Hospital’s Rule states that . . . (k) The Intermediate Value Theorem states that . . . (l) The graph of a function on an interval I is concave up if . . . 1.

. . . if f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N .

2.

. . . if f is a function that satisfies the following hypotheses: (i) f is continuous on the closed interval [a, b]. (ii) f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that f ′ (c) =

f (a + h) − f (a) if this limit exists. h h→0

. . . f ′ (a) = lim

3. 4.

f (b) − f (a) . b−a

. . . If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c, d ∈ [a, b].

5.

. . . the graph lies above all of its tangents.

6.

. . . is in the domain of f such that either f ′ (c) = 0 or f ′ (c) does not exist.

7.

. . . if lim f (x) = f (a) .

8.

x→a

. . . on a continuous curve where the curve changes from concave upward to concave downward or from concave downward to concave upward.

9.

. . . the base of the exponential function which has a tangent line of slope 1 at (0, 1).

10.

. . . If f (x) ≤ g(x) ≤ h(x) and lim f (x) = lim h(x) = L then lim g(x) = L. x→a

x→a

x→a

BA7A853D-157E-496E-8EEF-F0D1229988C3 final-exam-707d8

MATH 150 D100 / MATH 151 D100

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7. Consider the curve given by the polar equation r = 1 + 3 cos (2θ), [2]

[2]

π (a) Find the cartesian coordinates for the point on the curve corresponding to θ = . 6

(b) One of the graphs below is the graph of r = 1 + 3 cos (2θ). Indicate which one by putting a checkmark in the box below the graph you choose. You do not need to provide justification for your choice.

✷

[4]

0 ≤ θ ≤ 2π.

✷

π (c) Find the slope of the tangent line to the curve where θ = . 6

✷

7D6536A3-3ABE-4F5B-9B20-03BA297B7ACA final-exam-707d8

MATH 150 D100 / MATH 151 D100 [8]

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8. Consider a rectangle inscribed in a semicircle of radius 5 as shown below. In this question you are to find the maximum perimeter of the rectangle that can be inscribed inside this semicircle. √ (The final answer you should get for the maximum perimeter is 10 5, provide full details for obtaining this answer.)

5

7CA09614-11F4-422E-BC12-1BEC08D44340 final-exam-707d8

MATH 150 D100 / MATH 151 D100

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9. The three parts of this question are unrelated. [2]

(a) Give an example of a function that is strictly increasing on its domain.

[3]

(b) Give an example of a function f and an interval [a, b] such that the conclusion of the Mean Value Theorem is NOT satisfied for f on this interval.

[3]

(c) The function f (x) = x3 + ax2 + bx has a local extrema of value 2 at x = 1. Determine whether this extrema is a local maximum or a local minimum. You must justify your choice.

6209CE00-AEB2-438B-88E1-DDB4C9DF1692 final-exam-707d8

MATH 150 D100 / MATH 151 D100

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10. In this question we investigate the positive solution of the equation 2x = cos x [3]

(a) Explain why you know the equation has at least one positive solution.

[3]

(b) Show that the equation has exactly one positive solution.

[3]

(c) Use Newton’s Method to approximate the solution of the equation by starting with x1 = 0 and finding x2 .

B7E9BBED-4FF5-4315-9170-FA8BD8C92D8D final-exam-707d8

MATH 150 D100 / MATH 151 D100

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11. Consider the parametric curve x = 3 cos (2t), y = sin (4t),

y 1

0 ≤ t ≤ π.

The graph of this curve is given in the diagram.

−3

−2

−1

0

1

2

−1 [2]

(a) Find the coordinates of the point on the curve corresponding to t = 0, and draw this point on the graph above.

[2]

(b) Find the coordinates of the point on the curve corresponding to t = π/12, and draw this point on the graph above.

[2]

(c) On the graph above draw arrows indicating the direction the curve is sketched as the t values increase from 0 to π .

[3]

(d) Find the tangent line to the curve at the point where t = π/12.

[2]

(e) State the intervals of t for which the curve is concave up, and the intervals of t for which the curve is concave down. (You can determine this information directly from the graph above, you don’t need to use the second derivative.)

3 x

D8F57910-1402-4E0B-A390-6AE9EC3C4744 final-exam-707d8

MATH 150 D100 / MATH 151 D100 [8]

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12. You are riding on a Ferris wheel of diameter 20 meters. The wheel is rotating at 2 revolutions per minute. How fast are you rising when you are at the point P in the diagram, that is you are 6 meters horizontally away from the vertical line passing through the centre of the wheel?

898303E1-5CDF-4753-87CE-1C3CFC17AE12 final-exam-707d8

MATH 150 D100 / MATH 151 D100 [0]

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13. [Bonus: 3 Points] If f and g are two functions for which f ′ (x) = g(x) and g ′ (x) = f (x) for all x, then show that f 2 (x) − g 2 (x) must be a constant....