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Exercises: Calculus 1
Duong Thanh PHAM
August 29, 2021
1 Functions, Limits and Continuity
1.1 Functions, domains, compositions
Exercise 1.1. Find a formula for the inverse of the function
f (x) = x^2 + 2x, x ≥ − 1.
Exercise 1.2. Let f (x) =
2014 − cos(x + 1)
(i) Find the domain and the range of f.
(ii) Find functions g, h so that f (x) = g(h(x)), x ∈ (−∞, ∞).
Exercise 1.3. Which of the following is the domain of the function y =
5 − x +
x^2 − 2?
(A) {x ≤ 5 }
(B) {−
2 ≤ x ≤
(C) {x ≤ −
2 or
2 ≤ x ≤ 5 }
(D) {x ≥ 5 or x ≤ −
Exercise 1.4. Find the domains of the functions.
(i) f (x) = (^3) xx − 1
(ii) f (x) = (^) x (^2 5) + 3x^ + 4x + 2
(iii) f (t) =
t + 3
t
(iv) g(u) = √u +
4 − u
(v) h(x) = √^1 x^2 − 5 x
Exercise 1.5. Find the domain of the range of the function
h(x) =
4 − x^2.
Exercise 1.6. The graph below (Figure 1) shows a function f with domain [1, 4] and range [0, 2]. Let g(x) = 2f (x + 1).
Figure 1: This figure is used in Exercise 1.
(i) State the domain and range of g(x).
(ii) Sketch the graph of g(x).
Exercise 1.7. Let C be a circle with radius 2 centred at the point (2, 0).
(i) Write an equation for the circle C.
(ii) Is curve C the graph of a function of x? Explain your answer.
(iii) Write parametric equations to traverse C once, in a clockwise direction, starting from the origin.
Exercise 1.8. Let C be a circle centered at (0, 2) with radius 2. Which of the following is the parametric equation of C, in the counter-clockwise direction, starting from the point (0, 4)?
(A) x = 2 + 2 cos θ, y = 2 sin θ for θ ∈ [0, 2 π],
(B) x = 2 cos(θ + π 2 ), y = 2 sin(θ + π 2 ) for θ ∈ [0, 2 π],
(C) x = 2 cos(θ + π 2 ), y = 2 + 2 sin(θ + π 2 ) for θ ∈ [0, 2 π],
(D) x = 2 cos(θ − π 2 ), y = 2 + 2 sin(θ − π 2 ) for θ ∈ [0, 2 π],
Exercise 1.9. Find an expression for the function whose graph is the given curve.
(i) The line segment joining the points (1, −3) and (5, 7).
(ii) The bottom half of the parabola x + (y − 1)^2 = 0.
(iii) The top half of the circle x^2 + (y − 2)^2 = 4. (iv) The two functions whose graphs are as in the following picture.
Exercise 1.10. Determine whether is even, odd, or neither.
(i) (^) x 2 x+ 1
(ii) x
2 x^4 + 1
(iii) (^) x x+ 1
(iv) x |x|
(v) 1 + 3x^2 − x^4
(vi) 1 + 3x^3 − x^5
(A) 1
(B) 0
(C) 13
(D) the limit does not exist
Exercise 1.17. If dxe is the smallest integer that is not smaller than x, then lim x→ 1 dxe is
(A) 1
(B) 0
(C) 2
(D) the limit does not exist
Exercise 1.18. For the function f whose graph is given, state the following.
(i) (^) xlim→ 2 f (x) (ii) (^) x→−lim 1 − f (x)
(iii) (^) x→−lim 1 + f (x)
(iv) (^) xlim→∞ f (x)
(v) (^) x→−∞lim f (x)
(vi) The equations of asymptotes.
Figure 2: This figure is used in Exercise 1.
Exercise 1.19. Evaluate
xlim→ π 2
(cos x) cos(tan x).
Exercise 1.20. (i) Evaluate the limit
xlim→ 2
√^6 −^ x^ −^2 3 − x − 1. (ii) By using the Squeeze Theorem, or otherwise, evaluate the limit
xlim→π(x^ −^ π) sin^ x −π π.
Exercise 1.21. (i) Use Squeeze theorem to evaluate the limit
xlim→ 0 x^ cos(ln^ |x|). (ii) Find F ′(1) where F (x) = xf (xf (x)), and f (1) = 1, f ′(1) = 2.
Exercise 1.22. Find the following limits:
(i) (^) xlim→ 4
√ 3 x + 4 − x x − 4
(ii) (^) xlim→ 0 x^2
sin^2017 x + 2018
(iii) (^) xlim→ 0 +
esin^1 x
x + sin x.
Exercise 1.23. Let
f (x) =
sin x − 1 if x 6 = π/ 2 2 if x = π/ 2
Which of the following statements, I, II, and III, are true?
(I) (^) x→limπ/ 2 f (x) exists
(II) f (π/2) exists (III) f is continuous at x = π/2. (A) only I
(B) only II
(C) I and II
(D) all of them
Exercise 1.24. Let f (x) = x
(^2) − 4 x + 3 x − 1. Choose the correct statement: (A) (^) xlim→ 1 f (x) = − 2 (B) x = 1 is a vertical asymptote. (C) (^) xlim→ 1 f (x) = −∞
(D) f (x) = x − 3 for all x.
Exercise 1.25. (i) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after t minutes (in grams per liter) is C(t) = (^) 200 +^30 t t.
(ii) What happens to the concentration as t → ∞?
Figure 5: This figure is used in Exercise 1.
Exercise 1.28. From the graph of g (see Figure 5), state the intervals on which g is continuous.
Exercise 1.29. Which of the following is true for the function f (x) given by
f (x) =
2 x − 1 if x < − 1 x^2 + 1 if − 1 ≤ x ≤ 1 x + 1 if x > 1. (i) f is continuous everywhere, (ii) f is continuous everywhere except at x = −1 and x = 1, (iii) f is continuous everywhere except at x = −1, (iv) f is continuous everywhere except at x = 1, (v) None of the above.
Explain your choice in details.
Exercise 1.30. Let
g(x) =
cos x if x < 0 0 if x = 0 1 − x^2 if x > 0. (i) Explain why g(x) is discontinuous at x = 0. (ii) Sketch the graph of g(x).
Exercise 1.31. (i) Find the value of the constant k so that the function
f (x) =
kx^2 if x ≤ 2 x + k if x > 2 is continuous on (−∞, ∞).
(ii) Find lim x→ 1
√x − 1 x^2 − 1.
Exercise 1.32. The radius of the earth is roughly 4000 miles, and an object located x miles from the center of the earth weighs w(x) lb, where
w(x) =
ax if 0 < x ≤ 4000 b x^2 if^ x >^4000 and a and b are positive constants.
(i) Show that w(x) is continuous on (0; ∞) if and only if
a = 4000 b 3
(ii) Find any horizontal asymptotes and sketch the graph of w(x).
1.4 The Intermediate Value Theorem
Exercise 1.33. Show that there are two positive real numbers c satisfying
sin c = c^ + 1 3.
Exercise 1.34. Prove that the equation
ln x = e−x. has at least one root.
Exercise 1.35. Prove that the equation
x^2016 + (^) 1 + x (^284) + cos (^2) x = 119, has at least two roots.
Exercise 1.36. Show that the equation
x^3 − 2015 x^2 + 2x + 3 = 0
has three distinct real roots.
Exercise 1.37. Show that the equation
x^3 − x sin x − 1 = x
x + 2
has a real root in the interval [0, 2].
(i) Find the values of a and b such that f (x) is defined and continuous everywhere. (ii) With the values of a and b found in part (i), find any horizontal asymptotes, and sketch the graph of f (x). (iii) Show that f (x) is not differentiable at x = 1.
Exercise 2.5. Show that the function f (x) = |x − 2 | is continuous but not differentiable at x = 2.
Exercise 2.6. The graph of f is given. State, with reasons, the numbers at which is not differentiable
Figure 6: Exercise 2.
2.2 Finding Derivatives
2.2.1 Direct differentiation
Exercise 2.7. Find the derivative of
f (x) =
sin x x.
Exercise 2.8. Let f (x) =
1 + x^3 ; g(x) = e−^2 x+1. Evaluate the derivatives f ′(x), g′(x), and d dx (f^ (g(x))). Exercise 2.9. Suppose f and g are two differentiable functions on (−∞, ∞) and f (1) = 5, f ′(1) = −3, g(1) = 1, g′(1) = −1. Find F ′(1) where
F (x) = f (g(2x − 1)).
Exercise 2.10. Differentiate the function
h(x) = e−x^2 sin 2x.
Exercise 2.11. (i) Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g′(5) = 4.
(ii) If the tangent line to y = f (x) at (4, 3) passes through the point (0, 2), find f (4) and f ′(4).
Exercise 2.12. Let f (x) = ln
( (^) x + 2 αx + 3
. If the tangent to the graph of f (x) at x = 1 has slope − 1 /15, the value of the constant α is:
(A) 12 (B) 1 (C) 2 (D) 3
Exercise 2.13. The approximate value of y = √4 + sin x at x = 0.12, obtained from the tangent to the graph at x = 0, is
(A) 2. 00 (B) 2. 03 (C) 2. 06 (D) 2. 12
Exercise 2.14. The radius of a sphere is measured as 3cm with possible error 0.1cm. If we use this measurement to compute the volume, then the approximation (using differentials) for the maximal error of the volume is
(A) 11. 3 (B) 11. 7 (C) 12. 1 (D) 33. 9
Exercise 2.15. Find the parabola with equation y = ax^2 +bx whose tangent line at (1, 1) has equation y = 3x − 2.
Exercise 2.16. For what values of a and b is the line 2x + y = b tangent to the parabola y = ax^2 when x = 2?
Exercise 2.17. Find the value of c such that the line y = 32 x + 6 is tangent to the curve y = c√x.
Exercise 2.18. Suppose that f (5) = 1, f ′(5) = 6, g(5) = −3, and g′(5) = 2. Find the following values
(i) (f g)′(5) (ii) (f /g)′(5) (iii) (g/f )′(5)
Exercise 2.19. Suppose that f (2) = −3, g(2) = 4, f ′(2) = −2, and g′(2) = 7. Find h′(2).
(i) h(x) = 5f (x) − 4 g(x)
(ii) h(x) = f (x)g(x)
(iii) h(x) = f g^ ((xx))
(iv) h(x) = (^) 1 +g( fx ()x)
Exercise 2.20. If f (x) = exg(x), where g(0) = 2 and g′(0) = 5, find f ′(0).
Exercise 2.21. If h(2) = 4 and h′(2) = −3, find
d dx
( (^) h(x) x
x=
Exercise 2.22. If f and g are the functions whose graphs are shown, let u(x) = f (x)g(x) and v(x) = f (x)/g(x).
Exercise 2.26. Use logarithmic differentiation to differentiate the function
y =
x
)ln x
Exercise 2.27. Find the derivative of y = (sin x)cos^ x, 0 < x < π.
Exercise 2.28. Let y = (ax)bx^ where a and b are positive constants. The value of y′(1) is:
(A) 1
(B) b ln a + 1
(C) b ln a + b
(D) b(ln a + a)
2.2.3 Implicit differentiation
Exercise 2.29. Let 4 x^2 + 2xy^3 − 5 y^2 = 0.
Find dy/dx in terms of x and y.
Exercise 2.30. Evaluate dydx and ddx^2 y 2 at the point (0; −2) on the curve 4x^2 + y^2 = 4.
Exercise 2.31. Let g be a differentiable function on (−∞, ∞). Assume that
g(x) + x sin(g(x)) = x^2 , x ∈ (−∞, ∞).
Find g′(0) and write the equation of the tangent line to the graph of g at the point (0, 0).
Exercise 2.32. Find the equation of the tangent line to the curve x^3 + y^3 = x − y + 6 at the point (2, 0).
Exercise 2.33. Find the equation of the tangent line to the curve
x sin y + x^3 = (x − 1)^2 + 1,
at the point (1, 0).
Exercise 2.34. Find the equation of the tangent line to the the graph of
x^2 y + y^4 = 4 + 2x
at (− 1 , 1).
Exercise 2.35. The equation of the tangent to the hyperbola x^2 − y^2 = 12 at the point (4, 2) on the curve is
(A) x + y = 3
(B) y = 2x
(C) y = 2x − 6
(D) x + 2y = 6
2.2.4 Derivative of inverse
Exercise 2.36. Let f (x) = x
9 x^8 + 1 and let g be the inverse of f. Find g′(1/2).
Exercise 2.37. Let f (x) = x^5 + 4x − 8. Find (f −^1 )′(−3).
Exercise 2.38. The figure below (Figure 9) shows the graphs of two piece-wise functions, both having domain [0, 4]. Evaluate the following, or explain why they do not exist: (f ◦g)(3), (f ◦g ◦g)(3), g−^1 (1), f ′(3), and (f −^1 )′(3).
Figure 9: This figure is used in Exercise 2.
Exercise 2.39. This is the graph of f ′(x). This is the graph of f ′(x). This is the graph of f ′(x). Sorry to keep repeating myself, but you’re going to be really unhappy if you misread the problem. This is the graph of f ′(x).
(i) Find f ′′(4). Explain your method. (ii) Which is larger, f (1) or f (2)? Explain your answer.
(iii) Suppose f (−1) = 2. Find (f −^1 )′(2).
2.3 L’Hˆopital’s rule
Exercise 2.40. Evaluate the limit
xlim→ 1
√ (^3) x − 1 ln(2x − 1). Exercise 2.41. Use L’Hˆopital’s rule to evaluate the limit
Exercise 2.48. Two carts, A and B, are connected by a rope 28 meters long that passes over a pulley P (see the figure below, Figure 11). The point Q is on the floor 12 meters directly beneath P. Cart A is being pulled away from Q at a speed of 7 m/s. How fast is cart B moving towards Q at the instant when cart A is 5 m from Q?
Figure 11: This figure is used in Exercise 2.
Exercise 2.49. A balloon is being filled with helium at the rate of 4 cm^3 /min. The rate, in square cm per minute, at which the surface area is increasing when the volume is^323 π cm^3 is
(A) 4π (B) 2 (C) 4 (D) 1
Exercise 2.50. Two ships are sailing along straight-line courses that intersect at right angles. Ship A is approaching the intersection point at a speed of 20 knots (nautical miles per hour). Ship B is approaching the intersection at 15 knots. At what rate is the distance between the ships changing when A is 5 nautical miles from the intersection point and B is 12 nautical miles from the intersection point?
2.5 Maximum and minimum
Exercise 2.51. Find the absolute maximum and absolute minimum (if any) of f (x) = 5 − (x + 2)^4 /^5 on (−∞, ∞).
Exercise 2.52. Find the absolute maximum and absolute minimum of the function
f (x) = 20 − 3 x − (^12) x , x ∈ [2, 4].
Exercise 2.53. Find the maximum value and the minimum value of the function
f (x) = x^1 /^2 − x^3 /^2 , x ∈ [0, 4].
Exercise 2.54. Find the absolute maximum and absolute minimum values of f on the given interval
f (x) = (^) x 2 x+ 1 , x ∈ [0, 2].
Exercise 2.55. A student wants to draw a rectangle inscribed in a semicircle of radius 8. If one side must be on the semicircle’s diameter, what is the area of the largest rectangle that the student can draw?
Exercise 2.56. Find the area of the largest rectangle that can be inscribed in the ellipse x
2 16 +^
y^2 9 = 1.
Exercise 2.57. A population of animals is infected with a disease. After t days, the percentage of the population infected is modelled by the function p(t) = 8t e−^12 t^ for 0 ≤ t ≤ 60. Find the maximum value of p and the time at which it occurs.
Exercise 2.58. The concentration of a drug t hours after being injected into the arm of a patient is given by C(t) = (^) t (^2) + 0^0.^5 t. 81 , t ≥ 0.
When does the maximum concentration occur?
Exercise 2.59. The isosceles triangle shown above (Figure 12) has height AQ of length 3 and base BC of length 2. A point P is placed along the line segment AQ. What is the minimum value of the sum of the distances from P to A, P to B, and P to C?
Figure 12: This figure is used in Exercise 2.
Exercise 2.60. The absolute minimum value and absolute maximum value of the function f (x) = 10 x 1 + x^2 on^ [0,^ 2]^ are, respectively (A) 0 and 5 (B) 0 and 4
(C) −5 and 5 (D) −5 and 4
Exercise 2.61. On the curve y = x + x^4 , the point (2, 4) is
x
y
w
h
3
2
w 92 + h 42 = 1
Exercise 2.68. Two vertical poles (one is of 10m length and the other is of 5m) and stay 10m away from each other (see the figure). Find the point M on the ground between the two poles so that the sum of distances from M to the two high endpoints of the two poles attains its minimum value.
A
B
C M D
10
m
5 m
x (^10) − x
` 1
` 2
Hint: We want to find absolute minimum of the function (x) = 1 (x)+2 (x), where 1 (x) =
x^2 + 100 and ` 2 (x) =
(10 − x)^2 + 25 due to the Pythagorean Formula.
Exercise 2.69. Find the points on the ellipse 4x^2 + y^2 = 4 that are farthest away from the point (1, 0).
x
y
x
(x, y^ y
)
(1, 0)
Hint: If the point (x, y) lies on the ellipse then we have 4 x^2 + y^2 = 4. The distance between the two points (x, y) and (1, 0) is given by d =
(x − 1)^2 + y^2.
2.6 Proving inequality by the Mean value theorem
Exercise 2.70. Show that ln x < x ∀x > 0.
Exercise 2.71. Show that (^) √ 1 + h < 1 +^12 h ∀h > 0.
Exercise 2.72. Show that eu^ ≥ 1 + u for all u ≥ 0.
Exercise 2.73. Show that ex^ < 1 + xex^ ∀x > 0.
Exercise 2.74. Use the mean value theorem to prove that 1 2
1 + x
1 + x < √x + 2 √^1 x , x > 0.
2.7 Others
Exercise 2.75. The figure shows the graphs of f , f ′, and f ′′. Identify each curve, and explain your choices.
Exercise 2.76. The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
Exercise 2.77. At x = 1 the curve y = x^5 − 3 x^3 + 1 is
