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# MTH 2610: Midterm 2 Practice 1

Problem 1

Evaluate $$\displaystyle e^{7\ln8}$$.

Problem 2

If $$\displaystyle f(x)=\ln\frac{19x+3}{5x+2}$$, determine $$f^{\prime}(0)$$.

Problem 3

Find the derivative of $$\displaystyle f(x)=7e^{-9x+8}$$.

Problem 4

If $$\displaystyle f(x)=7^{ 8x+2}$$, determine $$f^{\prime}(0)$$.

Problem 5

Find the derivative of the function $$\displaystyle f(x)=x^{\ln\left(x^{6}\right)}$$.

Problem 6

If $$\displaystyle y^5-\ln\left(x^2+9\right)y-7x^3=17$$, determine $$\frac{dy}{dx}$$.

Problem 7

Find the derivatives of the following functions: (a) $$\displaystyle g(x)=\sin\left(x^3\right)$$; $$\quad$$ (b) $$\displaystyle f(x)=\frac{x^{8}+8^x}{\ln x}$$.

Problem 8

A ladder $$13$$ feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of $$4$$ feet per second. How fast is the top of the ladder moving down the wall when its base is $$12$$ feet from the wall?

Problem 9

• (a) Assume that $$A$$ and $$B$$ are differentiable functions such that $$B(3)=7$$, $$B^{\prime}(3)=6$$, $$A(7)=4$$, and $$A^{\prime}(7)=5$$. Let $$G$$ be the function defined as $$G(x)=A(B(x))$$. Determine $$G^{\prime}(3)$$.

• (b) Assume that $$P$$ and $$Q$$ are differentiable functions such that $$P(3)=7$$, $$P^{\prime}(3)=6$$, $$Q(3)=4$$, and $$Q^{\prime}(3)=3$$. Let $$H$$ be the function defined as $$H(x)=P(x)\cdot Q(x)$$. Determine $$H^{\prime}(3)$$.

• (c) Assume that $$f$$ is a differentiable function such that $$f^{\prime}(3)=6$$. Determine $$\displaystyle \lim_{h\to 0}\frac{f\left(\ln\left(e^{3}+h\right)\right)-f\left(3\right)}{h}$$.

Problem 10

The function $$y$$ depends on $$x$$ and is given implicitly in the following way: $y^y+x^x=x^{7}y^{3}+x^{4}.$ It is known that $$y(1)=1$$. Determine $$y^{\prime}(1)$$.

Problem 11

Determine and classify the critical values of the function $$\displaystyle f(x)= \sqrt{\left(x^2-13x+42\right)e^{2x+20}}$$ for $$-\infty< x< +\infty$$.