Calculator Training Tasks - Casio FX-991DE X

Practice Problems for Sections 1-5

Part 1: Mathematical Foundations

Problem 1.1: Fraction Calculations (x)

Use your calculator to evaluate the following expressions. Verify your answers are in simplified form.

  1. \(\frac{3}{4} + \frac{5}{6} - \frac{1}{2}\)

  2. \(\frac{7}{12} \times \frac{8}{21}\)

  3. \(2\frac{1}{3} + 1\frac{3}{4}\)

  4. \(\frac{15}{8} \div \frac{5}{12}\)

Problem 1.2: Powers and Roots (x)

Calculate the following using your calculator:

  1. \(3^4 + 2^5 - 5^2\)

  2. \(\sqrt{169} + \sqrt[3]{125}\)

  3. \((2^3)^2 - 4^3\)

  4. \(\sqrt[4]{256} \times \sqrt{49}\)

Problem 1.3: Logarithm Evaluations (x)

Use your calculator to find:

  1. \(\log_{10}(100)\)

  2. \(\ln(e^3)\)

  3. \(\log_2(64)\)

  4. \(\log_5(125) + \log_3(27)\)

Problem 1.4: Memory and Variables (x)

Follow these steps and record each result:

  1. Store the value 12.5 in variable A. Then calculate \(3 \times A + 7\).

  2. Calculate \(\sqrt{50}\) and store the result in B. Then find \(B^2\).

  3. If A = 5 and B = 3, use stored variables to calculate \(A^2 - B^2\).

Problem 1.5: Percentage Calculations (x)

Use your calculator’s percentage function to solve:

  1. What is 15% of 240?

  2. Increase 180 by 12%

  3. Decrease 500 by 8%

  4. A product costs 85€ after a 15% discount. What was the original price?

Problem 1.6: Prime Factorization (x)

Use the FACT function (SHIFT + FORMAT → Primfakt.) to find:

  1. Prime factorization of 360

  2. Prime factorization of 1260

  3. Use prime factorization to simplify \(\frac{84}{126}\)

Problem 1.7: GCD and LCM (x)

Use the GCD and LCM functions:

  1. Find GCD(48, 72)

  2. Find LCM(12, 18)

  3. Use GCD to simplify \(\frac{144}{180}\)

  4. Find the smallest number divisible by both 15 and 20

Problem 1.8: CALC Button - Formula Evaluation (x)

Use the CALC button to evaluate expressions with different variable values:

  1. Enter the quadratic formula \(\frac{-B + \sqrt{B^2 - 4AC}}{2A}\) and find one root of \(2x^2 + 5x - 3 = 0\)

  2. Enter the area formula \(A = \pi r^2\) and calculate for r = 3, r = 5, and r = 7

  3. Evaluate \(2x^2 + 3x - 5\) for x = 1, 2, 3, 4

Problem 1.9: Ans Button - Chain Calculations (x)

Use the Ans button to perform chain calculations:

  1. Calculate compound interest: Start with 1000€ at 5% annual interest. Use Ans to find the value after 1, 2, 3, and 4 years.

  2. Heron’s method for \(\sqrt{5}\): Start with guess 2, then iterate using \(\frac{1}{2}(x + \frac{5}{x})\)

Part 2: Equations & Problem-Solving

Problem 2.1: Quadratic Equations (x)

Use the polynomial equation solver to find the solutions of:

  1. \(x^2 - 7x + 12 = 0\)

  2. \(2x^2 + 5x - 3 = 0\)

  3. \(x^2 - 6x + 9 = 0\)

  4. \(x^2 + 4 = 0\) (What happens?)

Problem 2.2: Cubic Equations (xx)

Solve using the polynomial solver:

  1. \(x^3 - 6x^2 + 11x - 6 = 0\)

  2. \(x^3 - 3x^2 - 4x + 12 = 0\)

Problem 2.3: Systems of Linear Equations (x)

Use the simultaneous equation solver:

  1. \[\begin{cases} 2x + 3y = 13 \\ x - y = 1 \end{cases}\]

  2. \[\begin{cases} 5x + 2y = 24 \\ 3x - 4y = -2 \end{cases}\]

  3. \[\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 2 \end{cases}\]

Problem 2.4: Break-Even Analysis (xx)

A company’s profit function is \(P(x) = -2x^2 + 40x - 150\) where \(x\) is in hundreds of units.

  1. Find the break-even points using the polynomial solver.

  2. Find the vertex of this parabola using \(x_v = -\frac{b}{2a}\) and then calculate \(P(x_v)\).

  3. Interpret the business meaning of both results.

Problem 2.5: Market Equilibrium (xx)

Given supply and demand functions:

  • Demand: \(Q_d = 200 - 4P\)
  • Supply: \(Q_s = 50 + P\)

Set up and solve the system to find equilibrium price and quantity.

Problem 2.6: SOLVE Function - Newton’s Method (xx)

Use the SOLVE function (SHIFT + CALC) to find solutions numerically:

  1. Find one solution to \(x^3 - 5x + 3 = 0\) (start with initial guess x = 1)

  2. Solve \(\sin(x) = 0.5\) for x in radians (start with x = 0.5)

  3. Find where the derivative of \(f(x) = x^3 - 6x\) equals zero

Part 3: Functions

Problem 3.1: Function Evaluation (x)

Define \(f(x) = 2x^2 - 3x + 1\) in your calculator and find:

  1. \(f(0)\)

  2. \(f(2)\)

  3. \(f(-1)\)

  4. \(f(0.5)\)

Problem 3.2: Value Tables (x)

Create a value table for \(f(x) = x^2 - 4x\) from \(x = -1\) to \(x = 5\) with step 1.

  1. Record all function values.

  2. At which \(x\)-values is \(f(x) = 0\)?

  3. Where does \(f(x)\) appear to have its minimum?

Problem 3.3: Composite Functions (xx)

Define \(f(x) = 2x + 1\) and \(g(x) = x^2\) in your calculator.

Calculate:

  1. \(f(g(2))\)

  2. \(g(f(2))\)

  3. \(f(g(3))\)

  4. \(g(f(0))\)

Part 4: Advanced Functions

Problem 4.1: Exponential Calculations (x)

Calculate:

  1. \(e^2\)

  2. \(e^{-1}\)

  3. \(10^{2.5}\)

  4. \(e^{\ln(5)}\)

Problem 4.2: Trigonometric Calculations (x)

Make sure your calculator is in degree mode (D indicator). Calculate:

  1. \(\sin(30°)\)

  2. \(\cos(60°)\)

  3. \(\tan(45°)\)

  4. \(\sin^{-1}(0.5)\) (What angle has sine = 0.5?)

Problem 4.3: Angle Conversions (x)

  1. Convert \(45°\) to radians.

  2. Convert \(\frac{\pi}{3}\) radians to degrees.

  3. Calculate \(\sin\left(\frac{\pi}{6}\right)\) (make sure to use radian mode or convert).

Part 5: Differential Calculus

Problem 5.1: Numerical Derivatives (x)

For \(f(x) = x^2 - 4x + 3\), use the calculator’s derivative function to find:

  1. \(f'(0)\)

  2. \(f'(2)\)

  3. \(f'(4)\)

  4. At what value of \(x\) is \(f'(x) = 0\)?

Problem 5.2: Verifying Critical Points (xx)

For \(f(x) = x^3 - 6x^2 + 9x\):

  1. Find \(f'(1)\) and \(f'(3)\) using your calculator.

  2. Are \(x = 1\) and \(x = 3\) critical points? How do you know?

  3. Use the polynomial solver to verify by solving \(f'(x) = 3x^2 - 12x + 9 = 0\).

Problem 5.3: Second Derivative and Concavity (xx)

For \(f(x) = x^3 - 3x^2\):

  1. Find \(f'(x)\) manually: it should be \(3x^2 - 6x\).

  2. Use the calculator to verify: calculate \(f'(0)\), \(f'(1)\), \(f'(2)\).

  3. Now calculate \(f''(1)\) by finding the derivative of \(f'(x) = 3x^2 - 6x\) at \(x = 1\).

  4. Is \(x = 1\) a local maximum or minimum? (Check sign of \(f''(1)\))

Problem 5.4: Rate of Change (xx)

Revenue as a function of time is given by \(R(t) = -t^2 + 20t + 100\) (in thousands of euros).

  1. Find the rate of change of revenue at \(t = 5\) months.

  2. At what time is revenue changing at a rate of zero?

  3. Is revenue increasing or decreasing at \(t = 15\)? By how much?

Problem 5.5: Tangent Line Verification (xxx)

For \(f(x) = x^3\) at the point \((2, 8)\):

  1. Find \(f'(2)\) using the calculator.

  2. The tangent line at this point is \(y = 12x - 16\). Verify this by checking that:

    • The slope equals \(f'(2)\)
    • The line passes through \((2, 8)\)

Problem 5.6: Comprehensive Optimization (xxx)

A company’s profit function is \(P(x) = -0.25x^2 + 15x - 80\) where \(x\) is units in hundreds.

Use your calculator to find:

  1. The break-even points (use polynomial solver for \(P(x) = 0\))

  2. The rate of change of profit at \(x = 20\)

  3. The production level that maximizes profit (where \(P'(x) = 0\))

  4. The maximum profit