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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 \( \displaystyle f(x)= \sqrt[3]{\left(x^2-13x+42\right)e^{2x+20}} \) for \( -\infty< x< +\infty \).