Department of Mathematics and Statistics Math 100 – Calculus 1
Final Exam – Winter 2021
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Examination – Winter 2021 MATH 100
1. Evaluate the limit. Use the symbol ∞ or −∞, where approprate. (a) lim x−1
x→−2+ x2(x + 2)
(b) lim sinθ θ→0 θ+tanθ
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√t2 +9−3 (c) lim 2
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t→0 t
2. If2x≤g(x)≤x4−x2+2forallx,findlimg(x). x→1
[2]
Examination – Winter 2021 MATH 100
3. Sketch the graph of a function f that satisfies all of the given properties. lim f(x)=4, lim f(x)=2, lim f(x)=2, f(3)=3, f(−2)=1
x→3+ x→3− x→−2
4. Find the derivative of the function using the limit process.
f(x) = 1 − 1 x
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5. If the tangent line to y = f(x) at (4,3) passes through the point (0,2), find f(4) and f′(4). [2]
6. Find the derivative of the function. Where possible write your answer in factored form using positive exponents.
(a) f(x)=x3 +cosx−tan2x+π2 [1] (b) f(x) = [x+(x+sec2 x)3]4 [2]
(c) f(θ)= sec2θ [3] 1+tan2θ
Examination – Winter 2021 MATH 100 Page 4
(d) f(x) = √ x
x4 + 4
[3]
7. If F(x) = f(3f(4f(x))) where f(0) = 0 and f′(0) = 2 find F′(0) where f is a differentiable function. [3]
8. Find the slope of the tangent line to the curve at the given point. [4]
ysin2x = xcos2y, π, π 24
Examination – Winter 2021 MATH 100 Page 5
9. A man walks along a straight path at a rate of 4 ft/s. A searchlight is located on the ground
20 ft from the path and is kept focus on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight? [6]
Examination – Winter 2021 MATH 100 Page 6
x3 x2(x2 + 3) 2x(3 − x2) 10. Letf(x)=x2+1,f′(x)= (x2+1)2 andf′′(x)= (x2+1)3 .
(a) Find the coordinates of all intercepts and equations of all asymptotes of the function f. [2]
(b) Find the intervals where f is increasing or decreasing. Find the coordinates of any extrema. [2]
(c) Find the intervals where f is concave up or concave down. Find the coordinates of any inflection points. [2]
(d) Use the above information to sketch the graph of f. [2]
Examination – Winter 2021 MATH 100 Page 7
11. Find the x-values of the coordinates on the ellipse 4×2 + y2 = 4 that are farthest away from
the point (1, 0). Make sure to include a sketch in your solution and verify your answer using
the second derivative test. [6]
Examination – Winter 2021 MATH 100 Page 8
12. Use differentials to approximate √5 31.
[3]
13. Calculate the definite integral by using the limit definition. Use right-hand endpoints in your
calculation. 3
(x2 +2)dx
0
[5]
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14. Consider the function f (x) = x2 + 1 on the closed interval [−1, 2].
(a) Determine whether the Mean Value Theorem for derivatives can be applied to f on the closed interval. If the Mean Value Theorem can be applied, find all values of c guaranteed
by the theorem. [3]
(b) Determine whether the Mean Value Theorem for integrals can be applied to f on the closed interval. If the Mean Value Theorem can be applied, find all values of c guaranteed
by the theorem. [3]
Examination – Winter 2021 MATH 100 Page 10
2√
15. Evaluate (x + 3) 4 − x2 dx by writing it as a sum of two integrals and evaluate one of −2
those two integrals using a common geometric formula. [5]
16. Find(f−1)′(a)wheref(x)=3×3+4×2+6x+5anda=5. [4]
Examination – Winter 2021
17. Evaluate the integral.
(a) (θ3 − sec2 θ + 2) dθ
2 e1/x (b) x2 dx
1
MATH 100
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(c) √ 2x dx 3x−1
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Examination – Winter 2021
18. Find the derivative.
(a) y = esin 2x + sin(e2x)
(b) y = x5 + 5x
MATH 100
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19. Use logarithmic differentiation to find the derivative of the function.
y= e−xcos2x x2 + x + 1
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[2]
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Examination – Winter 2021
20. Solve the differential equation.
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dy = x2y dx