Problem Set 7.pdf

8/19/2019 Problem Set 7.pdf

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MH1100/MTH112: Calculus I.

Problem Set for the Week 8 Lectures. (Tutorial held during Week 9.)

This week’s topics:

• Diff erentiation rules.

• The calculus of the trigonometric functions.

• The chain rule.

The tutor will aim to discuss: 2, 8, 9, 12, 15, 25, 26, 31-35 and maybe other

problems.Problems that I think might be more unusual or difficult than normal aremarked with a ⋆.

In this problem set, unless otherwise indicated, you may use the rulesfor diff erentiation.

Problem 1: (#2.3.21 from [Stewart].)

Diff erentiate the function u(t) = 5√ t + 4

√ t5.


Diff erentiate the function v(x) =√

x + 13√ x

2.


Diff erentiate the function F (x) = x4−5x3+√ x

x2 in two diff erent ways: by using

the quotient rule, and by simplifying the expression first. Check you get thesame answer.


Diff erentiate the functionf (x) =

x

x + cx

.

1




Find the equation of the tangent line and the normal line to the curve

y =

√ x

x + 1

at the point (4, 0.4).


If f (x) is a diff erentiable function, find an expression for the derivative of each of the following functions:

(a) y = x2

f (x) (b) y =

f (x)

x2

(c) y = x2

f (x) (d) y = 1+xf (x)√ x


Show that the curve y = 6x3 + 5x− 3 has no tangent line whose slope is 4.


Find equations of the tangent lines to the curve y = x−1x+1 that are parallel

to the line x− 2y = 2.


Find a degree 2 polynomial P (x) such that P (2) = 5, P ′(2) = 3, P ′′(2) = 2.

Problem 10⋆: (#2.3.98 from [Stewart].)

Consider a function F (x) = f (x)g(x), where f and g are assumed to havederivatives of all orders f (n)(x) and g(n)(x).

(i) Show the following formula

F ′′(x) = f ′′(x)g(x) + 2f ′(x)g′(x) + f (x)g′′(x).

(ii) Find analogous formulas for F ′′′(x) and F (4)(x).

(iii)⋆⋆ Guess a formula for the case of general n, F (n)(x), then use inductionto prove your guess is correct.

2




A tangent line is drawn to the hyperbola xy = c at a point P .

(a) Show that the midpoint of the line segment cut from this tangent lineby the co-ordinate axes is P .

(b) Show that the triangle formed by the tangent line and the co-ordinateaxes always has the same area, no matter what point P is chosen.


Using standard facts, find a simple evaluation of the limit limx→1 x1000−1

x−1 .


Using your intuition, guess if there is a straight line that is tangent to bothof the parabolas y = x2 and y = x2− 2x + 2. Then decide if this line exists,and then find it, using equations.


Find the points on the curve y = cosx2+sinx

at which the tangent line is hori-zontal.


Determine the limit

limt→0

tan6t

sin2t .


Determine the limitlimθ→0

sin θ

θ + tan θ.

3




Determine the limit

limx→1

sin(x− 1)

x2 + x − 2.


Consider the functionf (x) =

x√ 1 − cos2x

.

Is f continuous at x = 0? If not, what type of discontinuity does it havethere? Justify your answer using standard properties of limits. It may beinteresting to check your answer with a graph of this function.

Problem 19⋆:

Determine the limit

limx→0

√ 1 + tanx−

√ 1 + sin x

x3 .

Problem 20⋆: (#10.27 from [Spivak].)

Let f be a function which is diff erentiable at 0, and assume that f (0) = 0.Prove that there exists a function g(x) which is continuous at 0 and suchthat

f (x) = xg(x).

Problem 21: (#2.5.5 from [Stewart])

Identify the following function as a composition h(x) = f (g(x)) then use thechain rule to diff erentiate it:

h(x) =√

sinx.

4




Identify the following function as a composition h(x) = f (g(x)) then use thechain rule to diff erentiate it:

h(x) =

4x− x2100

.


Diff erentiate the following function

f (x) = sin (x cosx) .



F (z) =

z − 1

z + 1.

5





y = sin

1 + x2.

Problem 26: (#2.5.42 and #2.5.43 from [Stewart])

Diff erentiate the functions:

(i) y = 1 +√ x.

(ii) y =

1 +

1 +√ x.


Find the tangent line to the curve

y = sin sinx

that goes through the point (π, 0).


Find the points on the graph of the following function

f (x) = 2 sinx + sin2 x

at which the tangent line is horizontal.


Suppose that f is diff

erentiable at every point of R

and that α ∈ R

. LetF (x) = f (xα) and let G(x) = [f (x)]α. Find expressions for:

(a) F ′(x), and

(b) G′(x).

6




If g is a twice-diff erentiable function, and f (x) = xg(x2), then find an ex-pression for f ′′(x) in terms of the derivatives of g(x).


If F (x) = f (3f (4f (x))), where f (0) = 0 and f ′(0) = 2, determine F ′(0).

Problem 32⋆: (#2.5.83 from [Stewart])

Use the chain rule to give proofs that:

(a) The derivative of an even function is an odd function.

(b) The derivative of an odd function is an even function.


Suppose that f (x) is a function which is diff erentiable everywhere and thaty = f (x) is a curve that always lies above the x-axis, and never has ahorizontal tangent. For what value of y is the rate of change of y5 with

respect to x eighty times the rate of change of y with respect to x?

Problem 34⋆: (#2.5.87 from [Stewart])

Use the chain rule to show that if θ is measured in degrees, then:

d

dθ(sin θ) =

π

180 cos θ.

Problem 35:

If it is known thatd

dx [f (2x)] = x2

then what is f ′(x)?

7

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Problem Set 7.pdf