New Zealand
Level 8 - NCEA Level 3

# Applications of differentiation

## Interactive practice questions

Consider the function $f\left(x\right)=3-e^{-x}$f(x)=3ex.

a

Determine $f\left(0\right)$f(0).

b

Determine $f'\left(0\right)$f(0).

c

Which of the following statements is true?

$f'\left(x\right)<0$f(x)<0 for $x\ge0$x0

A

$f'\left(x\right)<0$f(x)<0 for all real $x$x.

B

$f'\left(x\right)>0$f(x)>0 for all real $x$x.

C

$f\left(x\right)>0$f(x)>0 for all real $x$x.

D

$f'\left(x\right)<0$f(x)<0 for $x\ge0$x0

A

$f'\left(x\right)<0$f(x)<0 for all real $x$x.

B

$f'\left(x\right)>0$f(x)>0 for all real $x$x.

C

$f\left(x\right)>0$f(x)>0 for all real $x$x.

D
d

Determine the value of $\lim_{x\to\infty}f\left(x\right)$limxf(x).

Easy
Approx 4 minutes

Find the equation of the tangent to the curve $f\left(x\right)=2e^x$f(x)=2ex at the point where it crosses the $y$y-axis.

Express the equation in the form $y=mx+c$y=mx+c.

Consider the curve $f\left(x\right)=e^x+ex$f(x)=ex+ex.

By filling in the gaps, complete the proof showing that the tangent to the curve at the point $\left(1,2e\right)$(1,2e) passes through the origin.

The tangent to the curve $f\left(x\right)=2.5e^x$f(x)=2.5ex at the point $x=7.5$x=7.5 is parallel to the tangent to the curve $g\left(x\right)=e^{2.5x}$g(x)=e2.5x at the point where $x=k$x=k.

Solve for $k$k.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems