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6.04 Applications of the differentiation of logarithmic functions

Interactive practice questions

Consider the graph of $y=\ln x$y=lnx.

Loading Graph...

a

Is the function increasing or decreasing?

Increasing

A

Decreasing

B
b

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

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
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
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=ax2.

A

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

B

Linear, of the form $y=ax$y=ax.

C

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

D
Easy
2min

We want to find the gradient of the curve $y=\ln\left(x^2+5\right)$y=ln(x2+5) at the point where $x=3$x=3.

Easy
1min

Consider the function $y=\ln ax$y=lnax, where $a$a is a constant and $a,x>0$a,x>0.

Easy
4min

Consider the function $y=\ln\left(-x\right)$y=ln(x).

Easy
3min
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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-6

applies appropriate differentiation methods to solve problems

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