11 Lecture need to Solve Problems On Calculus 12.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.1, 3.2, 3.3There are 72 problems in total

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Section 2.2 / The derivative as a function

Overview

Last section, we defined the derivative (slope) of a function at a point. It

takes a little bit of computation to find a slope. Suppose we want to find the

slope of y = x3 at x = 0, 1, 2, 4, 7, 8, 12, 13, and 18? Rather than go through

the process of computing the limit for each of these numbers, we can compute

the limit at an arbitrary (unspecified) number, and then plug in whichever

specific number we like later. This leads to the definition of the derivative of

a function, which is itself a function.

Details

def: The derivative of the function f (x) is the function defined by

f (x + h) − f (x)

.

h→0

h

f 0 (x) = lim

rk: The domain of f 0 (x) is the set of all x for which the above limit exists.

Sometimes this is the same as the domain of f (x), but sometimes it is smaller.

discuss:

Given the graph of y = f (x), can you sketch the graph of

0

y = f (x)? How about vice versa?

ex: Find the derivative of f (x) =

√

x. Graph f and f 0 on the same axes.

ex: Find the derivative of

f (x) =

1−x

.

2+x

def: As stated in section 2.1, f is called differentiable at a if f 0 (a) exists.

Otherwise we say f is not differentiable at a.

def: We say f is differentiable on (a, b) if f is differentiable at each number

in the interval (a, b).

th: If f is differentiable at a, then f is continuous at a.

1

discuss: What does the graph of a function look like at a point where the

function is not differentiable?

discuss: Higher order derivatives – rate of change of the slope. Since the

derivative is just another function, you can take the derivative of a derivative.

The second derivative of position with respect to time is called acceleration.

discuss: The units of a derivative are the units of the function of the dependant variable per unit of the independent variable. For example, if position

is measured in feet and time is measured in seconds, then the first derivative

(velocity) is measured in feet per second. The second derivative would be

measured in feet per second per second.

Key vocab, techniques

vocab: derivative (of a function), differentiable (at a point or on an interval), differentiable / not differentiable, acceleration

techniques: Given a function, find its derivative (by using the limit definition of the derivative). Find the equation of the tangent line to a function at

a given number. Given the graph of a function, determine the graph of its

derivative.

Problems

1. Let f (x) =

√

1 − x.

(a) Use the limit definition of the derivative to find f 0 (x). Do not use

shortcut formulas.

(b) Give the domain of f (x) and f 0 (x).

(c) Find the equation of the tangent line to y = f (x) at x = −3.

√

2. Let f (x) = 2x + 1.

(a) Use the limit definition of the derivative to find f 0 (x). Do not use

shortcut formulas.

(b) Give the domain of f (x) and f 0 (x).

(c) Find the equation of the tangent line to y = f (x) at x = 4.

2

3. Let f (x) =

√

x + 5.

(a) Use the limit definition of a derivative to find f 0 (x). Please do not use

shortcut formulas.

(b) Find the equation of the tangent line to y = f (x) at x = −1.

√

4. Let f (x) = 1 − 2x.

(a) Use the limit definition of the derivative to find f 0 (x). Do not use

shortcut formulas.

(b) Give the domain of f (x) and f 0 (x).

(c) Find the equation of the tangent line to y = f (x) at x = −4.

5. Let f (x) =

1

.

x2 +1

(a) Use the limit definition of a derivative to find f 0 (x). Do not use shortcut

formulas.

(b) Give the domain of f (x) and f 0 (x).

(c) Find the equation of the tangent line to y = f (x) at x = −1.

6. Let f (x) =

√1 .

x

(a) Find f 0 (x) using the limit definition of the derivative. Do not use

shortcut formulas.

(b) Find the equation of the line tangent to y = f (x) at x = 4.

7. Use the limit definition of a derivative to find the derivative:

d

x3

dx

8. Use the limit definition of a derivative to find the derivative:

d 2

(x − x)

dx

3

Section 2.3 / Differentiation formulas

Overview

Having introduced the concept of a derivative of a function, we’d like to be

able to compute derivatives as easily as possible. Remember the limit laws?

Noticing some facts about limits of some simple functions, and how those

functions combine, let us take limits of more complicated functions. We’ll

do the same thing for derivatives in this section.

Details

fact:

•

d

(c)

dx

= 0 for any constant c

•

d

(x)

dx

=1

•

d

(cf )

dx

d

= c dx

f for any constant c

•

d

(xr )

dx

= rxr−1 for any real number r (the power rule)

•

d

(f

dx

•

d

(f g)

dx

•

0

f

g

± g) =

=

d

f

dx

±

d

g

dx

(the addition rule)

= f 0 g + g 0 f (the product rule)

f 0 g−g 0 f

g2

(the quotient rule)

ex: Find the derivatives of:

• f (x) = x4 − 3×2 + 4

√

• x x

√

• x x+1

•

1−x

2+x

•

x2 +x−2

x3 +6

Key vocab, techniques

vocab: power rule, product rule, quotient rule

techniques: Apply the differentiation formulas to find derivatives of functions.

Problems

1

1. Find the derivative: f (x) =

2. Using the facts that

where

d

dx

1

.

x2 +1

sin x = cos x and

d

dx

cos x = − sin x, find f 0 (x)

√

√

4

3

x2 tan x − x3 x4

f (x) =

.

sin(2x)

3. Find the equation of the line tangent to f (x) =

x

1+x2

at x = 1.

√

4.

Find the point on the graph y = x x with slope 3. Then find the

equation of the line tangent to the graph at that point.

5. Let f (x) = 13 x3 + 3×2 + 10x − 7.

(a) Show that f 0 (x) > 0 for every real number x.

(b) Argue that the graph y = f (x) is therefore always increasing.

(c) Conclude that f (x) has exactly one real zero.

2

Section 2.4 / Derivatives of trig functions

Overview

We find the derivatives of the six trigonometric functions.

Details

discuss: Now that we know what a graph of a function ought to look

like next to the graph of its derivative, let’s draw the graphs y = sin x and

y = cos x. Let’s then try hard to notice some things.

fact:

•

d

dx

sin x = cos x

•

d

dx

csc x = − csc x cot x

•

d

dx

cos x = − sin x

•

d

dx

sec x = sec x tan x

•

d

dx

tan x = sec2 x

•

d

dx

cot x = − csc2 x

Key vocab, techniques

vocab: none

techniques: Apply the derivatives of the trigonometric functions in any

type of problem dealing with differentiation we’ve seen thus far.

Problems

1. Find a point P on the graph of f (x) = tan x such that the tangent line

to y = f (x) at P is parallel to the line 4x − 3y = 2.

1

Section 2.5 / The chain rule

Overview

We’ve seen how to differentiate some basic types of functions. One thing

we don’t know how to handle

yet is compositions. For example, what is the

√

2

derivative of f (x) = sin x + 1? It would be extremely difficult to use the

limit definition of a derivative to solve this problem. We learn the chain rule,

which teaches us how to differentiate compositions.

Details

discuss: A prerequisite skill: recognizing a composition of functions, and

breaking the composition into its components.

discuss: Substitution in differentiation using Leibniz notation.

th: (the chain rule) If h(x) = f (g(x)), then h0 (x) = f 0 (g(x))g 0 (x). In

Leibniz notation, this looks like

d(f (g(x))) d(g(x))

d(h(x))

=

·

dx

d(g(x))

dx

ex: Some examples of differentiating compositions of functions.

Key vocab, techniques

vocab: chain rule

techniques: Use the chain rule to differentiate compositions of functions.

Given a table of values for two functions f and g and their derivatives, use the

chain rule to answer questions about the values of the derivatives of f (g(x))

and g(f (x)).

Problems

1.

Let h(x) = f (g(x)). Use the following table to order from least to

greatest h0 (2), h0 (3), h0 (5), h0 (7), h0 (11).

1

x f (x)

2

3

3

5

5

7

7

11

11

2

f 0 (x)

5

7

11

2

3

g(x) g 0 (x)

7

11

11

2

2

3

3

5

5

7

2.

Let h(x) = f (g(x)). Use the following table to order from least to

greatest h0 (2), h0 (3), h0 (5), h0 (6), h0 (0).

x f (x)

6

5

3

6

0

3

2

0

5

2

f 0 (x) g(x) g 0 (x)

3

2

0

5

0

2

2

5

6

3

5

6

6

3

0

3.

Let h(x) = f (g(x)). Use the following table to order from least to

greatest f (0), g(1), h0 (2), h0 (3), h0 (4).

f 0 (x) g(x) g 0 (x)

2

3

4

3

4

0

4

0

1

0

1

2

1

2

0

p

4. Find the derivative of f (x) = tan( x2 − sin2 x).

5. Find the derivative of f (x) = 4 sec tan sin(x2 + 1) .

x f (x)

0

1

1

2

2

3

3

4

4

0

√

6. Compute the derivative of f (x) = sin x2 + x − 2.

7. Compute the derivative of f (x) =

8. Find the derivative:

p

3

sec3 (x − 1) + x3 .

d

sin3 x

dx

9. Find the derivative:

d

dx

√

2

cos sin cos x − x

2

10. Find the derivative:

d

dx

r

x+

q

√

x+ x

q

11. Find the derivative of f (x) = 3 sec

12. Find the derivative of f (x) =

√

3

x2 −1

2+cos x

.

1

.

x3 +sin x

13. Find the derivative of f (x) = sin4 (x4 ).

14. Find the derivative of f (x) = sin4 (sin x4 ).

15. Find the derivative of f (x) = 4 sin tan sec(x2 + 1) .

16. Let f (x) =

2

.

sin(3x)+2

(a) Find the smallest positive number c such that f 0 (c) = 0.

(b) Using your answer from part (a), find the equation of the line tangent

to the graph y = f (x) at the point (c, f (c)).

3

Section 2.6 / Implicit differentiation

Overview

Consider the equation of the circle x2 + y 2 = 1. We can view y as a function

of x, so it makes sense to talk about the derivative y 0 . However, if we want

to find the derivative, using the techniques available up til now we would

need to solve for y first. In some equations it is inconvenient or impossible

to solve for y. That won’t stop us from finding y 0 as long as we understand

implicit differentiation.

Details

discuss: Suppose y is a function of x. The chain rule applies to functions

of y like so:

d(f (y)) d(y)

d

f (y) =

·

.

dx

d(y)

dx

Exploitation of this fact is known as “implicit differentiation.”

rk: Often, when using implicit differentiation, the resulting formula for y 0

will involve both x and y. If you wish, you can solve for y in the original

equation, and substitute it in.

ex: Find y 0 : x2 + y 2 = 25.

rk: Sometimes it’s not possible to solve for y. We can still find y 0 .

ex: x3 + y 3 = 6xy.

√

√

ex: 2 x + y = 3

ex: If f (x) + x2 (f (x))3 = 10 and f (1) = 2, find f 0 (1).

ex: Find the equation of the tangent line to the hyperbola

the point (x0 , y0 ).

Key vocab, techniques

vocab: implicit differentiation

1

x2

a2

−

y2

b2

= 1 at

techniques: Perform implicit differentiation in order to find derivatives of

functions without needing to solve for y. Apply this to all sorts of problems

involving finding derivatives (finding equations of tangent lines, finding where

a graph is increasing, etc).

Problems

1. Find the tangent line to the graph of x2 + y 2 = 5×4 at the point (1, 2).

2. Find the equation of the tangent line to the graph y = (x5 −31)32 at x = 2.

3. Find the tangent line to the graph of x + (xy)3 = 12 at 4, 21 .

4. Find the equation of the tangent line to x cos(xy) =

π

√

2

at (π, 41 ).

5. Find the derivative of y 2 (y 2 − 4) = x2 (x2 − 5) at the point (0, −2).

6. Find the derivative of 2(x2 + y 2 )2 = 25(x2 − y 2 ) at (3, 1).

√

7. Find the derivative of x2/3 + y 2/3 = 4 at (−3 3, 1).

8. Find the derivative of x2 + y 2 = (2×2 + 2y 2 − x)2 at 0, 21 .

2

…

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