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Calculus 30 - Final ReviewUnit One Introduction to Calculus1. Solve and express the answer using set and interval notation.
9 < 5 7x < 26 4
2. Use a sign analysis of factors to solve the following inequality. Specify the solution using set and interval notation.
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3. Find the equation of the piecewise function whose graph is shown.
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4. Evaluate the limit of the following indeterminate quotient.
5. Evaluate the limit of the following indeterminate quotient.
6. By using onesided limits, determine whether the limit of the following indeterminate quotient exists. Illustrate the results by sketching the graph.
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7. Evaluate the limit by making the suggested change of variable.
8. Evaluate the limit if it exists using any appropriate technique.
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Unit Two Differentiation
9. Find the equation of the tangent line to the curve at the point [2, f(2)]. Leave your answer in standard form.
10. Differentiate using the Product Rule and simplify.
s = (t3 + 1)(3 – 2t2)
11. Differentiate and express your answer in a simplified factored form.
f(x) = (x2 + 3)3(x3 + 3)2
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12. Find dy/dx.
a)
b)
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13. Find the rate of change of the function at the given value of t.
14. Use the Chain Rule to find dy/dx at the indicated value of x.
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Unit Three Applications of the Derivative
15. Find dy/dx using implicit differentiation.
y(x2 + 3) = y4 + 1
16. For the given curve, find the equation of the tangent line at the given point.
x2 + y2 = 1, (8,3)100 25
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17. A cyclist traveling east at 30 km/h has passed through an intersection. An observer, 56 m south of the intersection, watches the cyclist. How is the distance between the cyclist and the observer changing when the cyclist is 33 m from the intersection?
18. If a ball is dropped from a height of 72 m above the ground, its height after t seconds is given by the function: h(t) = 72 – 4.9t2.a) Find the average velocity of the ball between t = 1 and t = 3.b) Find the (instantaneous) velocity of the ball after 3s.
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19. A woman 2 m tall walks away from a streetlight that is 6 m high at the rate of 1.5 m/s.
a) At what rate is her shadow lengthening when she is 3 m from the base of the light?b) At what rate is her shadow lengthening when she is 30 m from the base of the light?
20. A box with an open top is to be made from a square piece of cardboard, of side length 100 cm, by cutting a square from each corner and then folding up the sides. Find the dimensions of the box of largest volume.
21. A man lives on an island 1 km from the mainland. His favourite pub is 3 km along the shore from the point on the shore closest to the island. The man can paddle his canoe at 3 km/h and can jog at 5 km/h. Determine where he should land so as to reach the pub in the shortest possible time.
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Unit Four Curve Sketching
22. Find the relative extrema and the points of inflection of the following polynomial function and then sketch the graph of the curve.f(x) = 4x3 + 18x2 + 3
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Unit Five Derivatives of Logarithm and Exponential Functions
23. Find the derivative of
24. Differentiate: p(u) = (e2u – e2u)2
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Unit Six Trigonometric Functions and Their Derivatives
25. Differentiate.
a) g(y) = sin πy cos πy b) y = (x + csc x)2
c) y = (cot x + sin x)2
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26. One end of a ladder of length 5 m slides down a vertical wall. When the upper end of the ladder is 3 m above the ground, it has a downward velocity of 0.5 m/s. Find the rate at which the angle of elevation θ is changing at that time.
27. A lighthouse searchlight 1 km from shore makes one revolution every 45 s. How fast is the spot of light moving along a wall on the shore when the spot is 500 m from the point P?
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Unit Seven Antiderivatives and Differential Equations
28. Find the general antiderivative of each function and verify your results by differentiation.
a) b) 2sin(πx) cos(πx)
c) d)10e5x 2e3x
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29. A pebble is dropped from the 15th floor of a skyscraper 45 m above the ground. At the same instant, a pebble is thrown downwards with a velocity of 20 m/s from the 20th floor 60 m above the ground. Which pebble strikes the ground first? Assume that the acceleration due to gravity is 10 m/s2 and neglect the effect of air resistance.
30. The number of bacteria in a culture increases at a rate proportional to the number present. If there are 1 000 bacteria at 1 pm and 1 200 at 3 pm, how many would there be at 8 pm?
31. A bowl of porridge initially at 80°C cools to 40°C in 15 minutes when the room temperature is 20°C. Jan refuses to eat her porridge if it cools to a temperature below 50°C. How long does she have before arriving at the table? Assume Newton’s law of temperature change.
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Unit Eight Areas and Integrals
32. Use the Fundamental Theorem to evaluate each definite integral. Verify the antiderivative by differentiation.
a)
b)
c)
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33. Let A be the area of the region bounded by the curve and the xaxis, over the given interval. Express Area as a definite integral and find its value using the Fundamental Theorem of Calculus.
34. Use u substitution to evaluate the following integrals.
a)
b)
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Click here tosee graph
35. Find the area between y = x2 – x and y = x over the interval –2 < x < 0. Draw a sketch showing the element of area.
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Click here to see graph
36. Find the total area enclosed by the curves y = x3 – x and y = x2 + x
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