Derivatives Exponential and Logarithmic Functions MCV4U TEST

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Q1. Simplify the following. Express all answers in exact form.

a. ln⁡(e^2x )-ln⁡(e^x ) b. 10^(-3 log⁡10 )
c. (x+y)^(log_2⁡π/log_2⁡(x+y) ) d. e^(-4 ln⁡2+2 ln⁡4 )
Q2. If log_2⁡P=x,log_2⁡Q=y,log_2⁡R=z, write log_2⁡((R^2 √Q)/P^3 ) in terms of x, y, and z.
Q3. Solve for x. Express in exact form.
a. log_2⁡∛((x^2+30x) )=2 b. log_4⁡〖x^3 〗+log_2⁡√x=8
Q4. Sketch the graph of y=log_2⁡(x+2)
Q5. Determine the derivative with respect to x. Simplify fully.
a. y=(1+5e^3x )^2 b. y=ln⁡(e^x √x)
c. f(x)=e/ln⁡x d. e^6x sin⁡(5x^2 )
Q6. Determine the equation of tangent line, in standard form, to the graph of the function f(x)=sin⁡(ln⁡〖x^2 〗 ) at x = -1.
Q7. Determine whether the graph of y=xe^(4x+1) is concave up or concave down when x = -1.
Q8. Suppose f(x)= log_2⁡(2x-1) find f^' (3).
Q9. Suppose f(x)=a ln⁡(2x+b) where f(e)=3 and f^' (e)=6/e , find the constants a and b.

[vanessawertheim]

first comment!

at question 6, i thought there was no solution (because i thought that x = -1 is not in the domain of lnx) but then i saw that the derivative of sin is cos, which is an even function so that's why the tangent at x = -1 exists!