11.16 Applications of the differentiation of logarithmic functions
Interactive practice questions
Consider the graph of $y=\ln x$y=lnx.
Is the function increasing or decreasing?
Increasing
Decreasing
Is the gradient to the curve negative at any point on the curve?
No
Yes
Which of the following best completes this sentence?
"As $x$x increases, the gradient of the tangent..."
decreases at a constant rate.
increases at an increasing rate.
increases at a constant rate.
decreases at an increasing rate.
increases at a decreasing rate.
decreases at a decreasing rate.
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.
decreases towards $-\infty$−∞.
decreases towards $0$0.
increases towards $\infty$∞.
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.
Exponential, of the form $y'=a^{-x}$y′=a−x.
Linear, of the form $y=ax$y=ax.
Hyperbolic, of the form $y'=\frac{a}{x}$y′=ax.
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.
Consider the function $y=\ln ax$y=lnax, where $a$a is a constant and $a,x>0$a,x>0.
Consider the function $y=\ln\left(-x\right)$y=ln(−x).