MATH1036
TUTORIAL 4
NOTES:
(i) The symbol ‘:=’ means we define the expression on the LHS to be the expression on the RHS.
(ii) The statement f : R → Z means that the function represented by the symbol f maps (takes) real values to
integer values.
(iii) The symbol ‘∀′ means ‘for all’ or ‘for every’. The symbol ∃ means ‘there exists’ (i.e., there is at least one).
Questions:
(1) Write down the derivatives of sin, cos, tan.
√
d
(2) dt cosec( e(tπ ) ) =
(3) Does the curve y = sin (x − sin x) have horizontal tangents at the x-axis? Give reasons.
(4) The “Lemniscate of Bernoulli” is a curve that looks like “∞”. Look up this curve on wikipedia to see a picture.
Consider the curve L, a lemniscate, with Cartesian equation:
(x2 + y 2 )2 = 2(x2 − y 2 ).
(a) A curve is a set of points in the plane each with an x-coordinate and a y coordinate that satisfy a
certain rule (here, for a (x, y) to be on the curve L, it must satisfy the equation of the curve). Write the
set L in set-builder notation. [Hint: this is not hard. The purpose of this question to demonstrate the
distinction between a curve – which is a set of points – and the equation of a curve which is the rule
that a point satisfies exactly when it is on the curve. Example: The parabola with the equation y = x2
is the set of points {(x, y) ∈ R2 : y = x2 }].
√ √
(b) Show that ( 23 , 12 ) ∈ L, i.e, show that ( 23 , 12 ) lies on the the curve L. [Hint: check that the coordinates
of this point satisfy the equation for L].
√ √
(c) Show that ( 12 , 23 ) ∈
/ L, i.e., show that ( 12 , 23 ) does not lie on the the curve L. [Hint: check that the
coordinates of this point do not satisfy the equation for L].
(d) By performing
√
implicit differentiation, determine the slope of the tangent line to the curve L at the
3 1
point ( 2 , 2 ) on L.
√
3 1
(e) Write down the equation for the tangent line to L at the point ( 2 , 2 ).
(5) Let p, q : R → R be functions that are everywhere differentiable, with ∀t ∈ R, q(t) ̸= 0. With s : R → R
3
defined as s(x) := e(p(x)) .q(x) , compute the derivative of s in terms of e, p, p′ , q, and q ′ . [Hint: use the chain
rule and product rule]
The following two questions are optional
(6) Use the Pythagorean theorem to compute cos(arcsin x) as a square root involving x ∈ ([−1, 1]). [Hint:
arcsin : [−1, 1] → [−π, π] is defined so that arcsin(x) is the unique value of the angle θ in radians (from the
√ x ∈ [−1, 1] and θ := arcsin(x), we have
interval [−π, π]) for which the value of sin θ equals x. Explicitly with
sin θ = x = x/1 and θ ∈ [−π, π]. Now: If sin θ = x/1, then cos θ = ?]
(7) Use the identity sin(arcsin x) = x and the chain rule to determine the derivative of arcsin in terms of a square
root involving x. [Hint: Use the previous problem]
1
TUTORIAL 4
NOTES:
(i) The symbol ‘:=’ means we define the expression on the LHS to be the expression on the RHS.
(ii) The statement f : R → Z means that the function represented by the symbol f maps (takes) real values to
integer values.
(iii) The symbol ‘∀′ means ‘for all’ or ‘for every’. The symbol ∃ means ‘there exists’ (i.e., there is at least one).
Questions:
(1) Write down the derivatives of sin, cos, tan.
√
d
(2) dt cosec( e(tπ ) ) =
(3) Does the curve y = sin (x − sin x) have horizontal tangents at the x-axis? Give reasons.
(4) The “Lemniscate of Bernoulli” is a curve that looks like “∞”. Look up this curve on wikipedia to see a picture.
Consider the curve L, a lemniscate, with Cartesian equation:
(x2 + y 2 )2 = 2(x2 − y 2 ).
(a) A curve is a set of points in the plane each with an x-coordinate and a y coordinate that satisfy a
certain rule (here, for a (x, y) to be on the curve L, it must satisfy the equation of the curve). Write the
set L in set-builder notation. [Hint: this is not hard. The purpose of this question to demonstrate the
distinction between a curve – which is a set of points – and the equation of a curve which is the rule
that a point satisfies exactly when it is on the curve. Example: The parabola with the equation y = x2
is the set of points {(x, y) ∈ R2 : y = x2 }].
√ √
(b) Show that ( 23 , 12 ) ∈ L, i.e, show that ( 23 , 12 ) lies on the the curve L. [Hint: check that the coordinates
of this point satisfy the equation for L].
√ √
(c) Show that ( 12 , 23 ) ∈
/ L, i.e., show that ( 12 , 23 ) does not lie on the the curve L. [Hint: check that the
coordinates of this point do not satisfy the equation for L].
(d) By performing
√
implicit differentiation, determine the slope of the tangent line to the curve L at the
3 1
point ( 2 , 2 ) on L.
√
3 1
(e) Write down the equation for the tangent line to L at the point ( 2 , 2 ).
(5) Let p, q : R → R be functions that are everywhere differentiable, with ∀t ∈ R, q(t) ̸= 0. With s : R → R
3
defined as s(x) := e(p(x)) .q(x) , compute the derivative of s in terms of e, p, p′ , q, and q ′ . [Hint: use the chain
rule and product rule]
The following two questions are optional
(6) Use the Pythagorean theorem to compute cos(arcsin x) as a square root involving x ∈ ([−1, 1]). [Hint:
arcsin : [−1, 1] → [−π, π] is defined so that arcsin(x) is the unique value of the angle θ in radians (from the
√ x ∈ [−1, 1] and θ := arcsin(x), we have
interval [−π, π]) for which the value of sin θ equals x. Explicitly with
sin θ = x = x/1 and θ ∈ [−π, π]. Now: If sin θ = x/1, then cos θ = ?]
(7) Use the identity sin(arcsin x) = x and the chain rule to determine the derivative of arcsin in terms of a square
root involving x. [Hint: Use the previous problem]
1
