Consider the graph of $y=\ln x$`y`=`l``n``x`.

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a

Is the function increasing or decreasing?

Increasing

A

Decreasing

B

Increasing

A

Decreasing

B

b

Is the gradient to the curve negative at any point on the curve?

No

A

Yes

B

No

A

Yes

B

c

Which of the following best completes this sentence?

"As $x$`x` increases, the gradient of the tangent..."

decreases at a constant rate.

A

increases at an increasing rate.

B

increases at a constant rate.

C

decreases at an increasing rate.

D

increases at a decreasing rate.

E

decreases at a decreasing rate.

F

decreases at a constant rate.

A

increases at an increasing rate.

B

increases at a constant rate.

C

decreases at an increasing rate.

D

increases at a decreasing rate.

E

decreases at a decreasing rate.

F

d

Which of the following best completes the sentence?

"As $x$`x` gets closer and closer to $0$0, the gradient of the tangent..."

increases towards a fixed value.

A

decreases towards $-\infty$−∞.

B

decreases towards $0$0.

C

increases towards $\infty$∞.

D

increases towards a fixed value.

A

decreases towards $-\infty$−∞.

B

decreases towards $0$0.

C

increases towards $\infty$∞.

D

e

We have found that the gradient function must be a strictly positive function, and it must also be a function that decreases at a decreasing rate. What kind of function could it be?

Quadratic, of the form $y'=ax^2$`y`′=`a``x`2.

A

Exponential, of the form $y'=a^{-x}$`y`′=`a`−`x`.

B

Linear, of the form $y=ax$`y`=`a``x`.

C

Hyperbolic, of the form $y'=\frac{a}{x}$`y`′=`a``x`.

D

Quadratic, of the form $y'=ax^2$`y`′=`a``x`2.

A

Exponential, of the form $y'=a^{-x}$`y`′=`a`−`x`.

B

Linear, of the form $y=ax$`y`=`a``x`.

C

Hyperbolic, of the form $y'=\frac{a}{x}$`y`′=`a``x`.

D

Easy

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